Math. Z. 232, 437–485 (1999) c Springer-Verlag 1999 Absolute continuity for curvature measures of convex sets II Daniel Hug Mathematisches Institut, Albert-Ludwigs-Universit¨ at, Eckerstr. 1, D-79104 Freiburg i. Br., Germany (e-mail: [email protected]) Received: January 8, 1998; in final form August 25, 1998 Mathematics Subject Classification (1991):52A20, 52A22, 53C65, 28A15 1. Introduction A central and challenging problem in geometry is to find the basic relation- ships between (suitably defined) curvatures of a geometric object and the local geometric shape of the object which is considered. In one direction, one asks for geometric properties of a set which can be retrieved, provided some specific information is available about the curvatures which are asso- ciated with the set. But it is also important to obtain inferences in the reverse direction. Here one wishes to find characteristic properties of the curvatures which can be deduced from knowledge of the local geometric shape of the sets involved. In convex geometry, where one strives to avoid a priori smoothness as- sumptions different from those already implied by convexity itself, curva- ture measures of arbitrary closed convex sets replace the pointwise defined curvature functions of smooth convex surfaces which are used in classical differential geometry. In spite of the lack of differentiability assumptions, (at least in principle) the curvature measures encapsulate all relevant infor- mation about the sets with which they are associated. In order to investigate these measures, the methods and tools of convex and integral geometry, certain generalized curvature functions and Federer’s coarea formula play a decisive rˆ ole. Our general framework is determined by the geometry of convex sets in Euclidean space R d (d ≥ 2). In this setting, local Steiner formulae are used to introduce the curvature measures C r (K, ·) of a (non-empty) closed convex
A central and challenging problem in geometry is to find the basic relation-ships between (suitably defined) curvatures of a geometric object and thelocal geometric shape of the object which is considered. In one direction,one asks for geometric properties of a set which can be retrieved, providedsome specific information is available about the curvatures which are asso-ciated with the set. But it is also important to obtain inferences in the reversedirection. Here one wishes to find characteristic properties of the curvatureswhich can be deduced from knowledge of the local geometric shape of thesets involved.
In convex geometry, where one strives to avoid a priori smoothness as-sumptions different from those already implied by convexity itself, curva-ture measures of arbitrary closed convex sets replace the pointwise definedcurvature functions of smooth convex surfaces which are used in classicaldifferential geometry. In spite of the lack of differentiability assumptions,(at least in principle) the curvature measures encapsulate all relevant infor-mation about the sets with which they are associated. In order to investigatethese measures, the methods and tools of convex and integral geometry,certain generalized curvature functions and Federer’s coarea formula play adecisive role.
Our general framework is determined by the geometry of convex sets inEuclidean spaceRd (d ≥ 2). In this setting,local Steiner formulaeare used tointroduce thecurvature measuresCr(K, ·) of a (non-empty) closed convex
438 D. Hug
setK ⊂ Rd, for r ∈ {0, . . . , d−1}, as Radon measures on theσ-algebra of
Borel subsets ofRd. These measures, as well as their spherical counterparts,the (intermediate)surface area measuresSr(K, ·), have been the subjectof numerous investigations over the last 30 years. This can be seen, e.g.,from the books of Schneider [41] and Schneider & Weil [44], which arerecommended for an introduction to this subject, as well as from the surveysby Schneider [42] and Schneider & Wieacker [46]. A considerable numberof these investigations can be understood as contributions to the followingfundamental question, which has also been pointed out in [43].
Which geometric consequences can be inferred for a closed convex setK,provided some specific measure theoretic information on the curvature mea-sureCr(K, ·), for somer ∈ {0, . . . , d−1}, is available? For example, whatcan be said about the set ofsingular boundary pointsof a closed convex setK if the singular part of some curvature measure ofK vanishes?
Of course, the curvature measures of special classes of convex bodies(non-empty compact convex sets) such as bodies with smooth boundaries(of differentiability classC2) or polytopes are fairly well understood. Forarbitrary closed convex sets, a systematic investigation was initiated in [24],which aims at establishing a precise connection between thelocal geometricshape, in particular theboundary structure, of a given convex setK and theabsolute continuityof some curvature measureCr(K, ·),r ∈ {0, . . . , d−2},of K with respect to theboundary measureCd−1(K, ·) of K (see Sect. 2for some definitions). There, based on the previous work [23], the interplaybetween the absolute continuity of some curvature measure of a convex setand the measure theoretic size of the set of singular boundary points of thisset has been elucidated. It is the purpose of the present paper to continuethis line of research.
One of the basic roots of the present research can be traced back to aresult of Aleksandrov. LetK ⊆ R
3 be a full-dimensional convex body, andsuppose that thespecific curvatureof K is bounded, that is, there is a constantλ ∈ R such thatC0(K, ·) ≤ λ C2(K, ·). ThenK is smooth(has a uniquesupport planethrough each boundary point); see [2] or [3, p. 445]. Obvi-ously, the assumption of bounded specific curvature precisely means that theGaussian curvature measureC0(K, ·) is absolutely continuous with respectto the boundary measureC2(K, ·) and the density function is bounded by aconstant. Aleksandrov’s result has been discussed in the books by Busemann[10, pp. 32–34] and Pogorelov [33, pp. 57–60] or in Schneider’s survey [38].These authors also raised the question whether suitable generalizations ofthis result could be established in higher dimensions. But only recently, anextension of Aleksandrov’s result to higher dimensions and all curvaturemeasures has been found by Burago & Kalinin [8]. As a consequence of
Curvature measures of convex sets II 439
their result, it follows that the assumption
Cr(K, ·) ≤ λ Cd−1(K, ·) , (1)
for a closed convex setK ⊂ Rd with non-empty interior, a constantλ ∈ R
andr ∈ {0, . . . , d − 2}, implies that the dimension of the normal cone ofK at an arbitrary boundary pointx is d− 1− r at the most. In the importantcase of the mean curvature measure, that is forr = d − 2, Bangert [6] andthe present author [25] have independently (and by different approaches)obtained a much stronger characterization, saying that condition (1) holds ifand only if a suitable ball rolls freely insideK. Thus it becomes apparent thatthe absolute continuity (with bounded density) of some curvature measureof a convex bodyK with respect to the boundary measure ofK allows oneto deduce a certain degree of regularity for the boundary surface ofK.
The much more restrictive assumption
Cr(K, ·) = λ Cd−1(K, ·) , (2)
for a convex bodyK ⊆ Rd with non-empty interior, a constantλ ∈ R and
r ∈ {0, . . . , d−2}, yields thatK must be a ball. This result, which was firstproved by Schneider [39], represents a substantial generalization of the clas-sical Liebmann-Suss theorem to the non-smooth setting of convex geometry.A different proof and extensions to spaces of constant curvature or to cer-tain combinations of curvature measures have been given by Kohlmann [29],[28]. For closed convex sets with non-empty interiors, Kohlmann (see [26],[27]) has also studied (weak) stability and splitting results under pinchingconditions of the form
α Cd−1(K, ·) ≤ Cr(K, ·) ≤ β Cd−1(K, ·) , (3)
whereα, β ∈ R are properly chosen constants. Furthermore, Bangert [6]has obtained an optimal splitting result in the caser = d−2. In some specialsituations, diameter bounds have been obtained; see, e.g., the contributionsby Diskant [13], Lang [30], and Bangert [6]. Conditions of the form (3)can be used to state stability results, which have been explored by variousauthors; see Diskant [12], Schneider [40], Arnold [4], Kohlmann [26], [27],and the literature cited there. Actually, in some of these papers argumentsare implicitly used which involve the absolute continuity of some curvaturemeasure. It is the purpose of the present paper to investigate the relationshipbetween the rather weak measure theoretic assumption of the absolute conti-nuity of some curvature measure and the geometry of the associated convexset. In particular, we are concerned withregularity results. Thus we alsoprovide the basis for subsequent work [25], in which the case of absolutecontinuity with bounded densities and some applications tostability resultsare treated.
440 D. Hug
For some of the results mentioned in the preceding paragraphs corre-sponding theorems are known for surface area measures. The degree ofsimilarity between statements of results and methods of proof for curva-ture and surface area measures depends on the particular case which isconsidered. For example, recent approaches to characterizations of balls orstability results for curvature measures differ from the proofs of correspond-ing results for surface area measures. Moreover, surface area measures aredistinguished by their connection to mixed volumes. Results for surfacearea measures which are in the spirit of the above mentioned theorems ofAleksandrov and Burago & Kalinin will be contained in [25] for the firsttime. There the interplay and analogy between surface area and curvaturemeasures is, in fact, exploited as a technique of proof. A careful analysis ofthe nature of this analogy suggests an underlyingduality, which will also bedescribed more precisely in [25]. As a prerequisite for this subsequent workand since the results are interesting in their own right, we shall establishresults concerning the absolute continuity of surface area measures whichare dual (in a vague sense) to those for curvature measures.
Acknowledgements.The author wishes to thank Professor Rolf Schneider for valuable com-ments on an earlier version of the manuscript. Thanks are also due to the referee for sugges-tions concerning the presentation of the paper.
2. Notation and statement of results
The starting point for the present investigation is Theorem 2.1 below. In orderto state it and to describe our main results, we fix some notation. LetCd bethe set of all non-empty closed convex setsK ⊂ R
d. LetHs, s ≥ 0, denotethes-dimensional Hausdorff measure in a Euclidean space. Which space ismeant, will be clear from the context. The unit sphere ofR
d with respect tothe Euclidean norm| · | is denoted bySd−1. If K ∈ Cd andx ∈ bdK (theboundary ofK), then thenormal coneof K atx is denoted byN(K, x); see[41] for notions of convex geometry which are not explicitly defined here.For our approach, the (generalized)unit normal bundleN (K) of a convexsetK ∈ Cd plays an important role. It is defined as the set of all pairs(x, u) ∈bdK × Sd−1 such thatu ∈ N(K, x). Walter (see [49] or [50]) showed thatthis set represents a (strong)(d − 1)-dimensional Lipschitz submanifold ofR
d×Rd. ForHd−1 almost all(x, u) ∈ N (K), one can introducegeneralized
curvatureski(x, u), i ∈ {1, . . . , d − 1}. These generalized curvatures canbe obtained as limits of curvatures which are defined on the boundaries ofthe outer parallel sets ofK. They are non-negative, sinceK is convex. Butthey are merely defined almost everywhere onN (K), since the boundariesof the outer parallel sets ofK are submanifolds which are of classC1,1, but
Curvature measures of convex sets II 441
need not be of classC2. More explicitly, for anyε > 0 letKε be the set of allz ∈ R
d whose distance fromK is at mostε. Fory ∈ bdKε letσK(y) denotethe exterior unit normal vector ofKε aty. Then, forHd−1 almost all(x, u) ∈N (K), the spherical image mapσK |bdKε is differentiable atx+ εu for allε > 0 (see [49]), and therefore curvaturesk1(x + εu), . . . , kd−1(x + εu)are defined as the eigenvalues of the symmetric linear mapDσK(x + εu)restricted to the orthogonal complement ofu. Hence, forHd−1 almost all(x, u) ∈ N (K) and anyε > 0, we can define
ki(x, u) := limt↓0
ki(x + εu)1 + (t − ε)ki(x + εu)
,
i ∈ {1, . . . , d−1}, independent of the particular choice ofε > 0 (see [53]).We shall always assume that the ordering of these curvatures is such that
0 ≤ k1(x, u) ≤ . . . ≤ kd−1(x, u) ≤ ∞ . (4)
In addition, we setk0(x, u) := 0 andkd(x, u) := ∞ for all (x, u) ∈ N (K).More details of this construction, in the more general context of sets withpositive reach, can be found in M. Zahle [53] and in [23], [24].
The curvature measures of a general convex setK cannot be expressed interms of curvature functions which are defined (almost everywhere) on theboundary ofK. However, the generalized curvature functions can be used todescribe curvature measures in an appropriate way. This is the reason why,for Hd−1 almost all(x, u) ∈ N (K), we define certain weighted elementarysymmetric functions of generalized curvatures onN (K) by
H j(K, (x, u)) :=(
d − 1j
)−1 ∑1≤i1<...<ij≤d−1
ki1(x, u) · · · kij (x, u)∏d−1i=1
√1 + ki(x, u)2
if j ∈ {1, . . . , d − 1}, and
H 0(K, (x, u)) :=d−1∏i=1
1√1 + ki(x, u)2
.
In the following, we refer to Chapter 1 of [14] for the basic notation andresults concerning measure theory. However, there is one minor difference.For us a Radon measure inR
d will be defined on the Borel subsets ofRd,
whereas in [14] Radon measures are understood to be outer measures definedon all subsets ofRd. The simple connection between these two points ofview is as follows. A Radon measureµ in the sense of [14] yields a Radonmeasure in our sense simply by restrictingµ to theσ-algebra of Borel sets.
442 D. Hug
On the other hand, a Radon measureµ on the Borel sets ofRd can beextended as a Radon measureµ to all subsets ofRd by setting
µ(A) := inf{
µ(B) : A ⊆ B , B ∈ B(Rd)}
.
Here and subsequently, we denote byB(X) theσ-algebra of Borel sets ofan arbitrary topological spaceX. The preceding discussion shows that wecan simply refer to Radon measures (onR
d) without further explanations.Similar remarks apply to Radon measures onSd−1.
Now let µ andν be two Radon measures onRd. If ν(A) = 0 implies
µ(A) = 0 for all A ∈ B(Rd), then we say thatµ is absolutely continuouswith respect toν, and we writeµ � ν. By the Radon-Nikodym theorem,µ � ν if and only if there is a non-negative Borel measurable functionf : R
d → R such that
µ(A) =∫
Af(x) ν(dx)
for all A ∈ B(Rd). In particular, thedensity functionf is locally integrablewith respect toν. Furthermore, we say thatµ is singularwith respect toνif there is a Borel setB ⊆ R
d such thatµ(Rd \ B) = 0 = ν(B), and in thiscase we writeµ ⊥ ν. Certainly, this is a symmetric relation. A version of theLebesgue decomposition theorem says that for arbitrary Radon measuresµandν there are two Radon measuresµa andµs such thatµ = µa + µs,µa � ν andµs ⊥ ν. Moreover, the absolutely continuous partµa and thesingular partµs (of µ with respect toν) are uniquely determined by theseconditions. We shall also consider the restriction(µ x A)(·) := µ(A ∩ ·) ofa Radon measureµ to a setA ∈ B(Rd), which is again a Radon measure.Similar definitions and statements apply to measures on the Borel subsetsof the unit sphere, where the surface area measures of convex bodies aredefined.
These notions and results will now be applied to the curvature measuresof a convex setK ∈ Cd. As these measures are locally finite and concentratedon bdK, the curvature measureCr(K, ·), for anyr ∈ {0, . . . , d − 1}, canbe written as the sum of two measures, that is,
Cr(K, ·) = Car (K, ·) + Cs
r (K, ·) ,
whereCar (K, ·) is absolutely continuous andCs
r (K, ·) is singular with re-spect to the boundary measureCd−1(K, ·). Recall that ifK ∈ Cd, then
Cd−1(K, ·) = Hd−1 x bdK
if K has non-empty interior or dimK ≤ d − 2. If dim K = d − 1, then
Cd−1(K, ·) = 2(Hd−1 x bdK) .
Curvature measures of convex sets II 443
Subsequently, we often say that ther-th curvature measure of a convex setis absolutely continuous, by which we wish to express that this measure isabsolutely continuous with respect to the boundary measure of the set.
The following result, which was proved in [24, Theorem 3.2], gives anexplicit description of the singular partCs
r (K, ·) in terms of the generalizedcurvature functions on the unit normal bundle ofK.
Theorem 2.1. For a convex setK ∈ Cd, r ∈ {0, . . . , d − 1}, and β ∈B(Rd),
Csr (K, β) =
∫N s(K)
1β(x)Hd−1−r(K, (x, u)) Hd−1(d(x, u)) (5)
if N s(K) is the set of all(x, u) ∈ N (K) such thatkd−1(x, u) = ∞.
In Sect. 3, we shall show how Theorem 2.1 can be used to prove a usefulcondition which is necessary and sufficient for the absolute continuity ofthe r-th curvature measure of a convex set. It is appropriate to state sucha characterization (Theorem 2.2) as a local result for curvature measureswhich are restricted to an arbitrary Borel subset ofR
d. Indeed, the abso-lute continuity of these measures merely depends on the local shape of theassociated convex set. The following theorem will also play a key role in[25].
Theorem 2.2. LetK ∈ Cd, r ∈ {0, . . . , d − 1}, andβ ∈ B(Rd). Then
Cr(K, ·) x β � Cd−1(K, ·) x β (6)
if and only if
kd−1(x, u) < ∞ or kr+1(x, u) = 0 or kr(x, u) = ∞ , (7)
for Hd−1 almost all(x, u) ∈ N (K) such thatx ∈ β.
It should be emphasized that condition (7) can be checked by simply count-ing the number of curvatures which satisfyki(x, u) = 0 or ki(x, u) = ∞,respectively. Also note that in the present situation condition (6) can beparaphrased by saying that the Radon measureCr(K, ·) on R
d is (d − 1)-rectifiable. This terminology is used, e.g., in [34, p. 603], [32, p. 228], or[16], where the(d − 1)-rectifiability of a general Radon measureµ is char-acterized in terms of properties of the(d − 1)-dimensional densities ofµ.However, these investigations do not seem to be directly related to the presentwork.
As defined in the introduction, a convex body is a non-empty compactconvex subset ofRd. LetKd denote the set of all convex bodies. In the specialbut important case of the curvature measureC0(K, ·) of a convex bodyK,
444 D. Hug
we obtain (again from Theorem 2.1) a characterization of absolute continuitywhich involves a spherical supporting property ofK. This property will bedescribed by using the setexpn∗K of directions of nearest boundary pointsof K. Formally, this is the set of all unit vectorsu ∈ Sd−1 for which thereexist pointsx ∈ intK andy ∈ bdK such that|y − x| = dist(x, bdK)andy − x = |y − x|u. In other words,u ∈ expn∗K if and only if a non-degenerate ball which is contained inK contains a boundary point ofK withexterior unit normal vectoru. In the following, we shall say thatK ∈ Kd issupported from inside by ad-dimensional ball in directionu if and only ifu ∈ expn∗K. Since we are dealing with a local result, we shall also need thespherical imageσ(K, β) of a convex bodyK at a setβ ⊆ R
d. Moreover,Dd−1h(K, u) denotes the product of the principal radii of curvature ofK atu. This product is defined forHd−1 almost allu ∈ Sd−1. It can be calculatedas the determinant of the Hessian of thesupport functionh(K, ·) of K ∈ Kd
restricted to the orthogonal complement ofu. For explicit definitions we referto [41].
Theorem 2.3. For a convex bodyK ∈ Kd andβ ∈ B(Rd), the followingthree conditions are equivalent:
(a) C0(K, ·) x β � Cd−1(K, ·) x β;(b) Dd−1h(K, u) > 0 for Hd−1 almost allu ∈ σ(K, β);(c) Hd−1(σ(K, β) \ expn∗K) = 0.
Statement (b) of Theorem 2.3 is an analytic and statement (c) a geometricway of characterizing the absolute continuity of the Gaussian curvaturemeasure. In fact, the geometric condition (c) can be viewed as a substantiallyweakened form of a condition requiring a suitable ball to roll freely insideK.
Using a Crofton intersection formulaand various integral-geometrictransformations, we extend Theorem 2.3 to curvature measures of any order.The corresponding result, Theorem 2.4, will be proved in Sect. 5. It can beinterpreted as a two-step procedure for verifying the absolute continuity ofcurvature measures of convex bodies with non-empty interiors. For the cur-vature measure of orderd − r of a convex bodyK andr ∈ {2, . . . , d − 1},the procedure essentially works as follows. First, one has to choose anr-dimensional affine subspaceE intersecting the interior ofK. Second, onehas to select a unit vectoru from the spherical image of the intersection
Curvature measures of convex sets II 445
K ∩ E and check whetheru ∈ expn∗(K ∩ E). The precise formulation in-volves the suitably normalized Haar measureµr on the homogeneous spaceA(d, r) of r-dimensional affine subspaces inR
d. Furthermore, here andin the following a prime which is attached to a quantity indicates that thisquantity has to be calculated with respect to an appropriate affine or linearsubspace. We denote the set of convex bodies with non-empty interiors byKd
o, andU(E) is the unique linear subspace which is parallel to a givenaffine subspaceE.
Theorem 2.4. LetK ∈ Kd0, β ∈ B(Rd), andr ∈ {2, . . . , d − 1}. Then
Cd−r(K, ·) x β � Cd−1(K, ·) x β
if and only if, forµr almost allE ∈ A(d, r) such thatE ∩ intK 6= ∅, theintersectionK ∩ E is supported from inside by anr-dimensional ball inHr−1 almost all directions of the setσ′(K ∩ E, β ∩ E) ⊆ U(E).
The main tool for establishing such an extension in Sect. 5 is the specialcases = r of the following theorem, which is of interest in its own right. Itrefers to the setCd
o of closed convex sets inCd with non-empty interiors.
Theorem 2.5. Let K ∈ Cdo, let β ∈ B(Rd), and assume thatr ∈ {2, . . . ,
d − 1} ands ∈ {r, . . . , d − 1}. Then
Cd−r(K, ·) x β � Cd−1(K, ·) x β
if and only if
C ′s−r(K ∩ E, ·) x (β ∩ E) � C ′
s−1(K ∩ E, ·) x (β ∩ E) ,
for µs almost allE ∈ A(d, s) such thatE ∩ intK 6= ∅.
Recall that the prime which is attached to the curvature measureC ′s−r(K ∩
E, ·) means that this measure has to be calculated with respect to the affinehull of K ∩E. Thus, fors = r, Theorem 2.5 especially says that in the meancurvature case (r = 2) absolute continuity can be verified by investigatingplanar sections ofK.
With regard to Theorem 2.4 it is natural to ask for a one-step procedurewhich allows one to decide whether a particular curvature measure of aconvex body is absolutely continuous with respect to the boundary measureor not. A result which leads to such a procedure is contained in the ensuingTheorem 2.6. It is based on the following definitions.
Let us fix a convex bodyK ∈ Kd and somer ∈ {0, . . . , d − 1}. Fora unit vectorv ∈ Sd−1 let H(K, v) denote the support plane ofK withexterior normal vectorv. An affine subspaceE ∈ A(d, r) is said totouchK if E ∩ K 6= ∅ andE ⊆ H(K, v) for somev ∈ Sd−1. Furthermore, wewriteA(K, d, r) for the((d−r)(r+1)−1)-rectifiable set ofr-dimensional
446 D. Hug
affine subspaces ofRd which touchK. OnA(K, d, r) several authors [51],[18], [54], [35], [44] have introduced naturally defined measures. For convexbodies, however, all these measures are essentially equivalent. These contactmeasures have been used for calculating collision probabilities [37], [52],and they are related to absolute or total curvature measures [36], [5], [45].Let us denote such a measure byµr(K, ·). Some relevant details will bedescribed in Sects. 4 and 5.
Next we define thespherical image of orderr of K ∈ Kdo atβ ∈ B(Rd)
for anyr ∈ {0, . . . , d − 1} by
σr(K, β) := {E ∈ A(K, d, r) : β ∩ bdK ∩ E 6= ∅} .
The caser = d−1 leads to the ordinary spherical image, sinceA(K, d, d−1)is the set of supporting hyperplanes ofK each of which can be identified withits exterior unit normal vector. Letωi denote the surface area of the(i − 1)-dimensional unit sphere. Then the measureµr(K, ·) will be normalized sothat the relation
Cd−1−r(K, β) =ωd
ωd−rµr(K, σr(K, β)) , (8)
due to Weil [51], holds for allβ ∈ B(Rd). Setu− := {tu : t ≤ 0} ifu ∈ R
d \ {o}, let r ∈ {2, . . . , d − 1}, and defineB(z, t) := {y ∈ Rd :
|y − z| ≤ t} if z ∈ Rd andt ≥ 0. Then we say thatK is supported from
inside by anr-dimensional ball atE ∈ A(K, d, r − 1) if there is somep ∈ K ∩ E, someu ∈ Sd−1 ∩ U(E)⊥ with (E + u−) ∩ intK 6= ∅, andsomeρ > 0 such thatB(p − ρu, ρ) ∩ (E + u−) ⊆ K.
Equation (8) provides an integral-geometric interpretation for curvaturemeasures of convex sets. In the present context, it also suggests a character-ization of absolute continuity involving touching planes.
Theorem 2.6. LetK ∈ Kdo, β ∈ B(Rd), andr ∈ {2, . . . , d − 1}. Then
Cd−r(K, ·) x β � Cd−1(K, ·) x β
if and only if K is supported from inside by anr-dimensional ball atµr−1(K, ·) almost allE ∈ σr−1(K, β).
Essentially, Theorem 2.6 is deduced from Theorem 2.4 through a successionof auxiliary results. The proof includes arguments from convexity, geometricmeasure theory and also some basic results about Haar measures. The keyidea is to associate with anr-dimensional affine subspaceE meeting intKand a unit vectoru ∈ U(E) the(r−1)-dimensional support plane ofK ∩Erelative toE with exterior unit normal vectoru. This support plane thenrepresents an(r − 1)-dimensional affine subspace which touchesK.
It has already become apparent that the boundary of a convex bodyK ∈ Kd
o one of whose curvature measures is absolutely continuous with
Curvature measures of convex sets II 447
respect to the boundary measure cannot be too irregular. A precise and ina certain sense optimal result in this spirit is stated as Theorem 4.6 in [24].Another regularity result, which complements the picture, is provided by thefollowing theorem. As usual, we say thatx ∈ bdK is a regular boundarypoint of K ∈ Kd
o if there exists precisely one support plane ofK passingthroughx.
Theorem 2.7. LetK ∈ Kdo, β ∈ B(Rd), r ∈ {2, . . . , d − 1}, and assume
that
Cd−r(K, ·) x β � Cd−1(K, ·) x β .
Then, forµr−1(K, ·) almost allE ∈ σr−1(K, β), every boundary point ofK which lies inE is regular.
In convex and integral geometry, the surface area measures are at leastas important as the curvature measures, and, perhaps, they are even morerelated to other parts of convexity. The surface area measuresSr(K, ·) aredefined for convex bodiesK ∈ Kd andr ∈ {0, . . . , d − 1} as measureson theσ-algebra of Borel subsets of the unit sphere. In addition,S0(K, ·)is equal to the restriction of the(d − 1)-dimensional Hausdorff measure totheσ-algebra of Borel subsets of the unit sphere. Therefore it is natural tostudy characterizations and implications of the condition
Sr(K, ·) x ω � S0(K, ·) x ω,
whereω ⊆ Sd−1 is an arbitrary Borel set. Indeed, for surface area mea-sures, we obtain results which are similar to those already described forcurvature measures. This will be shown in Sects. 3 and 4. In fact, a com-parison of results suggests an underlying duality which will be investigatedmore thoroughly in a subsequent paper [25].
3. Characterization of absolute continuity
We have already stressed the point that Theorem 2.1 from the introductionand, similarly, Theorem 3.5 from [24] (see also the proof of Theorem 3.6 inthis section) provide explicit expressions for the singular parts of the curva-ture and surface area measures of suitable convex sets, respectively. Theseexpressions now lead to a first characterization of the absolute continuity forcurvature and surface area measures, in terms of generalized curvature func-tions, if they are combined with Lemma 3.1 below. From these expressions,we can also deduce more geometric characterizations of absolute continuityin the special cases of the Gaussian curvature measure and the surface areameasure of orderd − 1. Note that by referring to absolute continuity wealways mean absolute continuity with respect to the boundary measure or
448 D. Hug
the surface area measure of order zero, that is, (in both cases) with respectto the suitably restricted(d − 1)-dimensional Hausdorff measure.
In this section, we shall first consider the case of curvature measures, andthen we discuss corresponding results for surface area measures. Section 4will exclusively be devoted to a thorough study of surface are measures,since for these measures the arguments seem to be slightly easier. Dualresults for curvature measures then constitute the subject of Sect. 5.
In the following, we refer to Schneider’s book [41] for notation andfor notions from convexity which are not defined here. From [14], [23]and [24] we adopt the terminology concerning measure theory. For ex-ample, normalized elementary symmetric functions of principal curvaturesHi(K, ·) or radii of curvatureDih(K, ·), for suitable convex setsK andi ∈ {0, . . . , d − 1}, are defined as in [24]. The conventions for calculationsinvolving ‘∞’ are the same as in [23,§2]. Further, in the caser = 0 theleft-hand side of Eq. (9) below is defined as
n∏j=1
√1 + a2
j
−1.
Of course, this is motivated by the expression by whichH 0(K, (x, u)) hasbeen defined.
Lemma 3.1. Let r, n ∈ N, 0 ≤ r ≤ n, n ≥ 1, anda1, . . . , an ∈ [0,∞].Assume thata1 ≤ . . . ≤ an. In addition, we definea0 := 0 andan+1 := ∞.Then ∑
1≤i1<...<ir≤n
ai1 · · · air∏nj=1
√1 + a2
j
= 0 (9)
if and only if eitheran−r+1 = 0 or an−r = ∞.
Proof. First of all, for arbitraryn ∈ N the special casesr = 0 andr = nare easily verified.
The general statement is proved by induction with respect ton ∈ N. LetA(n) be the statement of the lemma. StatementA(1) has already been provedby considering the special casesr = 0 andr = n. Hence, we assume thatA(n − 1) has been proved for somen ≥ 2. We show thatA(n) is true. Thecasesr = 0 andr = n have already been checked. Thus let1 ≤ r ≤ n − 1.Then the condition ∑
1≤i1<...<ir≤n
ai1 · · · air∏nj=1
√1 + a2
j
= 0 (10)
will be considered in each of the two casesan = ∞ and0 ≤ an < ∞.
Curvature measures of convex sets II 449
If an = ∞, then Eq. (10) is equivalent to∑1≤i1<...<ir−1≤n−1
ai1 · · · air−1∏n−1j=1
√1 + a2
j
= 0 , (11)
since all summands in Eq. (10) vanish which correspond to indices1 ≤ i1 <. . . < ir < n and since
an√1 + a2
n
= 1 .
Here, 0 ≤ r − 1 ≤ n − 1, n − 1 ≥ 1, a1, . . . , an−1 ∈ [0,∞], anda1 ≤ . . . ≤ an−1. SinceA(n − 1) is assumed to be true, Eq. (11) isequivalent toan−1−(r−1)+1 = 0 or an−1−(r−1) = ∞, that is,an−r+1 = 0or an−r = ∞.
If 0 ≤ an < ∞, then Eq. (10) implies thatan−r+1 · · · an∏n
This shows that Eq. (10) is equivalent toan = ∞ and(an−r+1 = 0 or an−r = ∞)
or0 ≤ an < ∞ andan−r+1 = 0 .
But this exactly is the statement ofA(n). utRecall that in order to simplify the presentation, we complemented the
definition of the generalized curvatures on the unit normal bundle of a convexsetK by settingk0(x, u) := 0 andkd(x, u) := ∞ for (x, u) ∈ N (K). Thiswill help us to avoid the need to distinguish different cases.
Proof of Theorem 2.2.By Theorem 2.1, the condition
Cr(K, ·) x β � Cd−1(K, ·) x β
is equivalent to∫N s(K)
1β(x)H d−1−r(K, (x, u)) Hd−1(d(x, u)) = 0 .
But this is tantamount to saying that forHd−1 almost all(x, u) ∈ N (K)such thatx ∈ β,
(x, u) /∈ N s(K) or H d−1−r(K, (x, u)) = 0 . (12)
450 D. Hug
Here and subsequently, we tacitly use the essential fact that the generalizedcurvature functions which are associated with convex sets are non-negative.From Lemma 3.1 and the definition of the setN s(K), we obtain that con-dition (12) is equivalent to
which was to be proved. utThe following two corollaries are designed to illustrate Theorem 2.2.
Corollary 3.2. LetK ∈ Cd, β ∈ B(Rd), andi ∈ {0, . . . , d − 1}. Then
Cr(K, ·) x β � Cd−1(K, ·) x β
for all r ∈ {i, . . . , d − 1} if and only if
kd−1(x, u) < ∞ or ki(x, u) = ∞ ,
for Hd−1 almost all(x, u) ∈ N (K) such thatx ∈ β.
Corollary 3.3. LetK ∈ Cd, β ∈ B(Rd), andi ∈ {0, . . . , d − 1}. Then
Cr(K, ·) x β � Cd−1(K, ·) x β
for all r ∈ {0, . . . , i} if and only if
kd−1(x, u) < ∞ or ki+1(x, u) = 0 ,
for Hd−1 almost all(x, u) ∈ N (K) such thatx ∈ β.
Theorem 2.2 also yields a sufficient condition for the absolute continuityof all curvature measures of a given convex setK. In the following corollary,the assumption on bdK∩β implies that the restriction of the spherical imagemapσK to the setβ is locally Lipschitzian. Recall that thespherical imagemapof a convex setK ∈ Cd
o is defined for regular boundary points, andfor such a boundary pointx it is equal to the unique exterior unit normalvector ofK atx; see [41,§2.2]. If, in addition, we assume that the Lipschitzconstant ofσK |(bdK ∩ β) is smaller than a constantc, then we obtain thatki(x) ≤ c, for i ∈ {1, . . . , d−1} andHd−1 almost allx ∈ bdK ∩β, whichyields bounds for the densities of the curvature measures ofK.
Corollary 3.4. Let K ∈ Cdo, β ∈ B(Rd), and assume thatbdK ∩ β
is locally of classC1,1. ThenCi(K, ·) x β � Cd−1(K, ·) x β for all i ∈{0, . . . , d − 2}.
Proof. Use, for example, Lemma 3.1 from [24] and Theorem 2.2. ut
Curvature measures of convex sets II 451
In the special case whereK is a convex body of revolution, Theorem2.2 can be used to establish a simple characteristic condition for the abso-lute continuity of the curvature measures ofK. We fix some notation. Let(e1, . . . , ed) be an orthonormal basis ofR
d. Let f : (a, b) → [0,∞) be aconcave function, and define the curve
γ : (a, b) → R2 , t 7→ ted + f(t)e1 .
The convex body which is obtained by rotatingγ around theed-axis andtaking the closed convex hull is denoted byK, andK ′ := K ∩ lin{e1, ed}.Furthermore, define the function
F : (a, b) × Sd−2 → Rd , (t, u) 7→ ted + f(t)u ,
whereSd−2 := Sd−1 ∩ lin{e1, . . . , ed−1}.
Theorem 3.5. Let α ∈ B((a, b)) and d ≥ 3. Then the following threeconditions are equivalent:
(a) C0(K ′, ·) x γ(α) � C1(K ′, ·) x γ(α);(b) Ci(K, ·) x F (α × Sd−2) � Cd−1(K, ·) x F (α × Sd−2) for all i ∈
{0, . . . , d − 2};(c) Ci(K, ·) x F (α × ω) � Cd−1(K, ·) x F (α × ω) for somei ∈ {0, . . . ,
d − 2} and someω ∈ B(Sd−2) with Hd−2(ω) > 0.
Proof. An elementary calculation yields that
N(t, u) := σK(F (t, u)) =u − f ′(t)ed√
1 + f ′(t)2, (t, u) ∈ (a, b) × Sd−2 ,
wheneverf is differentiable att. If, in addition,f isC1 on(a, b) and secondorder differentiable att, then it follows that
∂N
∂t(t, u) = − f ′′(t)√
1 + f ′(t)23
∂F
∂t(t, u)
and
∂N
∂ui(t, u) =
1f(t)
√1 + f ′(t)2
∂F
∂ui(t, u) , i ∈ {1, . . . , d − 2} ,
whereu ∈ Sd−2 and(u1, . . . , ud−2, u) is an orthonormal basis of the sub-space lin{e1, . . . , ed−1}. If these relations are applied to the parallel bodiesof K andK ′, respectively, then one can see thatk(y, v) is defined for some(y, v) ∈ N (K ′) with y ∈ γ(α) if and only if k1(y, v), . . . , kd−1(y, v) aredefined for(y, v) ∈ N (K). Moreover, if one of these conditions is fulfilled,thenk1(y′, v′), . . . , kd−1(y′, v′) are also defined, whenever(y′, v′) is ob-tained from(y, v) by rotation around theed-axis. Note that we denote by
452 D. Hug
k(·) the curvature function on the unit normal bundle ofK ′ and that the as-sumption (4) on the ordering of the generalized curvatures ofK is suspendedduring the present proof. Nevertheless, Theorem 2.2 remains applicable ifinterpreted properly. Furthermore, if one of the previous conditions is ful-filled, then we also have
rem 2.2.First, (a) is fulfilled if and only ifk(y, v) < ∞ for H1 almost all(y, v) ∈
N (K ′) such thaty ∈ γ(α). But then we obtain from (13), (14) and a Fubini-type argument thatki(y, v) < ∞ holds for alli ∈ {1, . . . , d − 1} and forHd−1 almost all(y, v) ∈ N (K) such thaty ∈ F (α × Sd−2). Therefore (b)is true.
Obviously, (b) implies (c).Finally, assume that (c) is fulfilled. Due to (14) this yields thatkd−1(y, v)
< ∞ must hold forHd−1 almost all(y, v) ∈ N (K) such that(y, v) ∈F (α×ω). If the set of all(y, v) ∈ N (K ′) such thaty ∈ γ(α) andk(y, v) =∞ has positiveH1 measure, then the set of(y, v) ∈ N (K) such thaty ∈F (α×ω) andkd−1(y, v) = ∞ has positiveHd−1 measure. This can be seenfrom Hd−2(ω) > 0 and a Fubini-type argument. This contradiction showsthatk(y, v) < ∞ for H1 almost all(y, v) ∈ N (K ′) such thaty ∈ γ(α),and hence (a) must be true. ut
To see an application of Theorem 3.5 concerning a question of boundaryregularity, assume thatCi(K, ·) x F (α × ω) � Cd−1(K, ·) x F (α × ω)holds for somei ∈ {0, . . . , d − 2}, an intervallα ⊆ (a, b), and someω ∈ B(Sd−2)withHd−2(ω) > 0. Hence we obtain thatC0(K ′, ·) x γ(α) �C1(K ′, ·) x γ(α), and this implies thatγ|α is of classC1. But thenF |(α ×Sd−2), too, is of classC1. This should be compared with the immediateconclusion which can be obtained from Theorem 4.6 in [24].
Now, we are going to prove Theorem 2.3. Recall from [21] or fromthe introduction the definition of the setexpn∗K of directions of nearestboundary points of a convex bodyK, which can be rewritten in the form
expn∗K ={
u ∈ Sd−1 : B(x, ρ(K − x, u)) ⊆ K for somex ∈ intK}
,
whereρ(K − x, ·) denotes theradial functionof K with respect tox.
Curvature measures of convex sets II 453
Proof of Theorem 2.3.The equivalence of (b) and (c) follows from Lemma2.7 in [21] if the second order differentiability almost everywhere of thesupport functionh(K, ·) is used; see [1] and the Notes for Chap. 2.5 in [41].
It remains to prove that (a)⇔ (b). But (a) is equivalent to∫N s(K)
1β(x)H d−1(K, (x, u))Hd−1(d(x, u)) = 0 . (15)
Recall that a unit vectoru ∈ Sd−1 is said to be aregular normal vectorofK if the support setF (K, u) of K with exterior normal vectoru consistsof a single point. This point is denoted byτK(u), and the correspondingmapτK , which is defined on the set of regular normal vectors, is called thereverse spherical image mapof K. It is known thatHd−1 almost all unitvectors are regular normal vectors of a given convex bodyK; see Theorem2.2.9 in [41].
An application of the coarea formula (Theorem 3.2.22 in [17]) to theprojection mapπ2 : R
This immediately follows from (16) and (17). Finally, note that
Ca0 (K, γ) = Hd−1(σ(K, γ)) − Cs
0(K, γ) ;
compare Eq. (4.2.21) in Schneider [41]. utRemark 1.It should be emphasized that even ifDd−1h(K, u) > 0 for Hd−1
almost allu ∈ Sd−1, the convex bodyK is not smooth in general. As acounterexample for dimensiond = 3 one can choose the polar bodyK∗of a suitable translate of the convex bodyK which is defined in Remark2 below. The fact that the conditionD2h(K∗, u) > 0 is fulfilled for H2
almost allu ∈ S2 can, for example, be deduced from Theorem 2.2 of [22].
454 D. Hug
For future investigations of the subject, it will be essential to have char-acterizations of absolute continuity for both curvature measures and surfacearea measures. Therefore the remaining part of this section is mainly devotedto briefly establishing results for surface area measures which, in a certainsense, are dual to those already obtained for curvature measures. We shallalso provide some explicit examples which can serve to illustrate the ab-stract results. But these examples also demonstrate that certain conclusionscannot be obtained without additional assumptions.
Theorem 3.6. LetK ∈ Kd, r ∈ {0, . . . , d − 1}, andω ∈ B(Sd−1). Then
Sr(K, ·) x ω � S0(K, ·) x ω
if and only if
k1(x, u) > 0 or kr+1(x, u) = 0 or kr(x, u) = ∞ ,
for Hd−1 almost all(x, u) ∈ N (K) such thatu ∈ ω.
Proof. LetNs(K) denote the set of all(x, u) ∈ N (K) such thatk1(x, u) =0. Then the assumption
Sr(K, ·) x ω � S0(K, ·) x ω
is equivalent to∫Ns(K)
1ω(u)H d−1−r(K, (x, u))Hd−1(d(x, u)) = 0 .
This is an immediate consequence of Theorem 3.5 in [24]. In other words,for Hd−1 almost all(x, u) ∈ N (K) such thatu ∈ ω,
(x, u) /∈ Ns(K) or H d−1−r(K, (x, u)) = 0 .
An application of Lemma 3.1 thus completes the proof. utAs in the case of the Gauss curvature measureC0(K, ·), the absolute
continuity of Sd−1(K, ·) can be characterized by a spherical supportingproperty. The situation here is ‘dual’ to the previous one. Condition (c) ofTheorem 3.7 below can be interpreted as a substantially weakened form ofa condition demandingK to roll freely inside a ball. The statement of thistheorem involves the setexp∗K of farthest boundary pointsof a convexbody K (see [21]). This definition implies thatx ∈ exp∗K holds if andonly if the boundary of a ball which containsK passes throughx. In thefollowing, we say thatK is supported from outside by ad-dimensional ballatx if and only ifx ∈ exp∗K. Moreover, recall from [41,§2.2] thatτ(K, ω)denotes thereverse spherical imageof K at a setω ⊆ Sd−1. By definition,τ(K, ω) is equal to the union of the support setsF (K, u) with u ∈ ω.
Curvature measures of convex sets II 455
Theorem 3.7. Let K ∈ Kd andω ∈ B(Sd−1). Then the following threeconditions are equivalent:
(a) Sd−1(K, ·) x ω � S0(K, ·) x ω;(b) Hd−1(K, x) > 0 for Hd−1 almost allx ∈ τ(K, ω);(c) Hd−1(τ(K, ω) \ exp∗K) = 0.
if σK denotes the spherical image map, which is defined forHd−1 almostall boundary points ofK. In addition, it follows from Lemma 3.1 in [24]that
k1(x, σK(x)) = 0 ⇐⇒ Hd−1(K, x) = 0 ,
for Hd−1 almost allx ∈ bdK. This finally implies the equivalence of (a)and (b).
For the proof of the additional statement note that due to Eq. (4.2.24)in [41], the relationSd−1(K, α) = Hd−1(τ(K, α)) holds for an arbitraryBorel setα ∈ B(Sd−1). utRemark 2.Even if Hd−1(K, x) > 0 is fulfilled for Hd−1 almost allx ∈bdK, it does not follow thatK is strictly convex ifd ≥ 3. The follow-ing counterexample is due to Dekster [11]. Denote by(e1, e2, e3) the stan-dard basis ofR3. Let K be the closure of the convex hull of the image setX((−1, 1) × (−π, π)), where
X : (−1, 1) × (−π, π) → R3 ,
(y, t) 7→ ((1 − y2) sin t, y, (1 − y2)(1 + cos t)) .
456 D. Hug
Then the segment[−e2, e2] is contained in the boundary ofK, althoughone can show thatH2(K, x) exists even in the sense of classical differentialgeometry and is positive for allx ∈ bdK \ [−e2, e2]. It should be empha-sized, however, that there is no positive constantc such thatH2(K, x) ≥ cis true forH2 almost allx ∈ bdK. Another example for a different purposeis given below.
It should also be observed that strict convexity does not imply thatHd−1(K, x) > 0 holds for a set of pointsx ∈ bdK which has positive(d − 1)-dimensional Hausdorff measure. This follows, for instance, from aBaire category argument.
Remark 3.Statements analogous (dual) to Corollaries 3.2–3.5 can be provedfor surface area measures as well.
In the following we shall describe the construction of a convex bodyK ∈K3
0 for whichS2(K, ·) is absolutely continuous with respect toS0(K, ·), butfor whichS1(K, ·) is not absolutely continuous. Also note that ifS1(K, ·) isabsolutely continuous, thenS2(K, ·) can still have point masses. An examplewill be given in [25].
Example 1.First of all we define three convex surfacesF1, F2, Fα3 by
The convexity ofF2 is clear, sinceF2 is obtained by rotating the strictlymonotone convex curve
x 7→(
x, 0, x +45x2)
, x ∈ [0, 1] ,
Curvature measures of convex sets II 457
around thee3-axis. The convexity ofF1 andFα4 can be proved with the help
of Tietze’s theorem; see, for example, Theorem 4.10 of Valentine’s book[48]. For the smooth boundary points ofF1 andFα
4 the local supportingproperty, which is required for the application of Tietze’s theorem, can bechecked by verifying that the Gauss-Kronecker curvature is positive. Forthe non-smooth boundary points the local supporting property can be seendirectly.
Now, let Eα be the unionE1 ∪ E2 ∪ Eα3 of the epigraphsE1, E2, Eα
3of F1, F2, Fα
3 , respectively. ThenEα is a closed convex set ifα ∈ (0, 1]is sufficiently small. To see this check the local supporting property for thepoints which belong to the curves
x 7→(
x, 0, |x| +45x2)
, x ∈ [−1, 1] ,
and
x 7→ (x, 1, |x| + x2) , x ∈ [−1, 1] .
In fact, it can be shown that anyα ∈ (0, 1] is suitable. But this requiressome calculations. Let us denote byE one such suitable set. Finally, setK1 := E ∩ H−(e3, 1), and letK2 be the reflection ofK1 at the hyperplaneH(e3, 1). Here,H−(e3, 1) := {x ∈ R
d : 〈x, e3〉 ≤ 1}, H(e3, 1) is thebounding affine hyperplane, and〈· , ·〉 denotes the Euclidean scalar product.ThenK := K1 ∪ K2 is the required convex body. For a visualization ofK, see Fig. 1 below* . Again convexity can be proved by verifying the localsupporting property for the points in bdK ∩ H(e3, 1).
The absolute continuity ofS2(K, ·) can be seen from Theorem 3.7, sinceK has been constructed in such a way thatH2(K, x) > 0 for H2 almostall x ∈ bdK. This follows from explicit calculations. The first surfacearea measure,S1(K, ·), however, is not absolutely continuous. To see this,consider the setN1 which is defined by
N1 := {((0, y, 0), u) ∈ N (K) : y ∈ (0, 1)} .
Again by construction we haveH2(N1) > 0, and forH2 almost allv ∈ N1we also have
k1(v) = 0 and k2(v) = ∞ ,
since the straight edge{(0, y, 0) ∈ R3 : y ∈ (0, 1)} consists of ridge
points of order one. Hence, the previous statement immediately followsfrom Theorem 3.6. Alternatively, this can be proved from the representationof the surface area measures as coefficients of a local Steiner formula.
* I am obliged to Dr. Alfred Schmidt for producing the figure with the GRAPE packagedeveloped at Bonn and Freiburg.
458 D. Hug
Fig. 1 Example of a convex bodyK ∈ K3o for which S2(K, ·) is absolutely continuous,
butS1(K, ·) is not absolutely continuous with respect toS0(K, ·)
A dual example for curvature measures follows by using the polar bodyof K with respect to a suitable choice of the origin. This kind of argumentis investigated more thoroughly in a subsequent paper [25].
We conclude this section by stating a sufficient condition for the absolutecontinuity of mixed surface area measures. In fact, Corollary 3.8 belowimproves a remark in Aleksandrov’s fundamental paper [1]. For a definitionof and results on mixed surface area measures we refer to Schneider [41].In addition, for a convex bodyK ∈ Kd, let Sd−1(K) be the set of all unitvectorsu ∈ Sd−1 for which there is a pointx ∈ bdK and someR > 0 suchthatK is contained in the closed ball of radiusR centred atx − Ru.
Corollary 3.8. Let Ki ∈ Kd, ω ∈ B(Sd−1), and suppose thatω ⊆Sd−1(Ki) for all i ∈ {1, . . . , d − 1}. ThenS(K1, . . . , Kd−1, ·) x ω �Hd−1 x ω.
Proof. Use Theorem 3.7 and Eq. (5.1.17) from Schneider [41]. ut
4. Integral-geometric results: surface area measures
The principal aim of this section is the derivation of an integral-geometricextension of Theorem 3.7, which treats the case of surface area measures
Curvature measures of convex sets II 459
of any order. Actually, we prove two such extensions in Theorems 4.5 and4.6. The basic idea underlying the proofs of these results is to use integral-geometricprojection formulaefor surface area measures. Such formulaerelate thei-th surface area measure of a convex body inR
d to thei-th sur-face area measure of projections ofK ontoj-dimensional subspaces (i < j)by averaging the latter with respect to a Haar measure on the Grassmannmanifold of j-dimensional linear subspaces ofR
d. The projection formu-lae and additional integral-geometric transformations (Lemmas 4.1 and 4.3)lead to an integral-geometric characterization of absolute continuity for sur-face area measures which is stated as Theorem 4.4. From this and Theorem3.7 we deduce Theorem 4.5. Variants and further applications of Theorem4.4 will be given in [25].
The second characterization, Theorem 4.6, is stated in terms of touchingaffine subspaces and supporting orthogonal spherical cylinders (definitionswill be given later in this section). This result can in turn be used to makeprecise (in Theorem 4.7) the intuitive feeling that a convex body one ofwhose surface area measures is rectifiable (absolutely continuous) shouldnot deviate too much from astrictly convexbody.
Before we can go further, some additional notation is needed. Forj ∈{1, . . . , d}, let G(d, j) := G(Rd, j) be the Grassmann manifold ofj-dimensional linear subspaces ofR
d, let
Gu(d, j) := {V ∈ G(d, j) : u ∈ V }if u ∈ Sd−1, and define the flag manifold
G0(d, j, 1) := {(u, V ) ∈ Sd−1 × G(d, j) : u ∈ V } .
It is well known thatG(d, j) andG0(d, j, 1) together with the natural oper-ation of the orthogonal groupO(d) of the Euclidean spaceRd are homoge-neousO(d)-spaces, and thatGu(d, j), for eachu ∈ Sd−1, is a homogeneousO(u⊥)-space with respect to the canonical operation of the subgroup
O(u⊥) := {ρ ∈ O(d) : ρu = u} .
Let us denote byνj , νuj , andνj, 1 the corresponding normalized Haar mea-
sures ofG(d, j), Gu(d, j), andG0(d, j, 1), respectively. Recall thatωj :=Hj−1(Sj−1), for j ∈ {1, . . . , d}, and denote byK|V the orthogonal pro-jection of the convex bodyK onto the linear subspaceV ∈ G(d, j). Fur-thermore, note thathK|V = hK |V if hK = h(K, ·) is the support function
of K ∈ Kd andhK|V is considered as a function defined onV ∈ G(d, j).Finally, observe that
DUi h(K|U, u) = det
(d2hK|U (u)|(u⊥ ∩ U)
),
460 D. Hug
i ∈ {1, . . . , d − 1}, holds for all(u, U) ∈ G0(d, i + 1, 1) for whichhK |Uis second order differentiable(sod) atu. We write DU
i h(L, ·) if L is aconvex body which is contained in thej-dimensional linear subspaceU andj ∈ {i + 1, . . . , d − 1}, in order to indicate that this expression has to becalculated with respect to the linear subspaceU . The same convention isused for surface area measures such asSU
i (L, ·).The next two lemmas are required to justify the repeated interchange of
the order of integration in the proof of Theorem 4.4 below.
Lemma 4.1. Let j ∈ {2, . . . , d − 1}, and letf : G0(d, j, 1) → [0,∞] beBorel measurable. Then
ωj
ωd
∫Sd−1
∫Gu(d, j)
f(u, V ) νuj (dV ) Hd−1(du)
=∫G(d, j)
∫Sd−1∩V
f(u, V ) Hj−1(du) νj(dV )
= ωj
∫G0(d, j, 1)
f(u, V ) νj, 1(d(u, V )) .
Proof. This can be proved in a similar way to Satz 6.1.1 in Schneider andWeil [44]. utLemma 4.2. LetK ∈ Kd, i ∈ {1, . . . , d−2}, andj ∈ {i+1, . . . , d−1}.Then the following three statements hold:
(1) D1 := {(u, V ) ∈ G0(d, j, 1) : hK|V is (sod) atu} is a Borel set;(2) (u, V ) 7→ DV
i h(K|V, u) is Borel measurable onD1;(3) νj, 1(G0(d, j, 1) \ D1) = 0.
Proof. Let {Lm : m ∈ N} be a dense set of linear functionals onRd, and
let {Bn : n ∈ N} be a dense set of bilinear functionals onRd × R
d. Form, k, l ∈ N, we defineWmkl as the set of all(u, V ) ∈ G0(d, j, 1) for whichthe implication
|x| <1k
⇒ |hK(u + x) − hK(u) − Lm(x)| ≤ 1l|x|
holds for allx ∈ V , and then we set
D0 :=⋂l∈N
⋃m, k∈N
Wmkl .
ThusD0 is equal to the set of all(u, V ) ∈ G0(d, j, 1) for which hK|V isdifferentiable atu. This implies thatD0 is a Borel set, sinceWmkl is a closedset.
Curvature measures of convex sets II 461
Furthermore, forn, k, l ∈ N, we defineUnkl as the set of all(u, V ) ∈ D0for which the implication
|x| <1k
⇒∣∣∣∣hK(u + x) − hK(u) − ⟨DV hK|V (u), x
⟩− 12Bn(x, x)
∣∣∣∣≤ 1
l|x|2
is true for allx ∈ V , and thus we obtain that
D1 =⋂l∈N
⋃n, k∈N
Unkl .
Now we can complete the proof as follows. ConsiderD0 as a topologicalsubspace ofG0(d, j, 1). In the subspace topology ofD0, the setUnkl isclosed, since the map
D0 → Rd , (u, V ) 7→ DV hK|V (u) ,
is continuous. But thenD1 ∈ B(D0) = D0 ∩B(G0(d, j, 1)); see [19, Satz1.2.10]. By the definition of the traceσ-algebraD0 ∩ B(G0(d, j, 1)), thiscompletes the proof, since we have already shown thatD0 ∈ B(G0(d, j, 1)).
The second statement is easy to see, and the third statement follows fromthe first one and from Lemma 4.1 if the second order differentiability almosteverywhere of a convex function is used. utRemark 4.By essentially the same proof it follows that
D2 := {(u, V ) ∈ G0(d, j, 1) : hK is (sod) atu}
is a Borel set. Although one has the obvious inclusionD2 ⊆ D1, it is stilltrue thatνj, 1(G0(d, j, 1) \ D2) = 0.
The following lemma expresses a result which is known in the specialcasej = i+1. For this case, it is mentioned without a proof in [9,§19.3.5],and, forj = i + 1 = d − 1, the recent paper by Barvinok [7], Lemma 2.3and Theorem 2.4, contains a proof which is different from the subsequentargument. Obviously, Lemma 4.3 can be extended to a relation betweenmixed discriminants by the usual method of polynomial expansion. It shouldalso be emphasized that Lemma 4.3 can be viewed as an algebraic version(for quadratic forms) of integral-geometric projection formulae for surfacearea measures. In fact, for convex bodies with support functions of classC2,the lemma is implied by such integral-geometric formulae. In the generalcase, we prefer to proceed in a different way.
462 D. Hug
Lemma 4.3. Let K ∈ Kd, i ∈ {1, . . . , d − 2}, j ∈ {i + 1, . . . , d − 1},and assume thathK is second order differentiable atu ∈ Sd−1. Then
Dih(K, u) =∫Gu(d, j)
DVi h(K|V, u) νu
j (dV ) .
Proof. First, let us assume thatj = i+1. Setud := u, and let(u1, . . . , ud)be an orthonormal basis ofR
d such thatu1, . . . , ud−1 are eigenvectors ofd2hK(u)|u⊥ with corresponding eigenvaluesr1, . . . , rd−1. Let (e1, . . . ,ed−1) denote the standard basis ofR
d−1. Then define the(d−1)× i-matrixP and the diagonal matrixD by
P := (e1 . . . ei) and D :=
√r1 O
...O
√rd−1
.
Finally, for ρ ∈ O(u⊥), set
Sρ :=(sρjl
)d−1
j, l=1:= (〈ρul, uj〉)d−1
j, l=1 .
Then we obtain that(d2hK(u)(ρul, ρuk)
)il, k=1 = P>S>
ρ D>DSρP ,
and from this we infer that
det((
d2hK(u)(ρul, ρuk))il, k=1
)=
∑1≤j1<...<ji≤d−1
rj1 · · · rji
[det((〈ujl
, ρuk〉)il, k=1
)]2.
Denote byνu the normalized Haar measure onO(u⊥). Then∫Gu(d, i+1)
The last integral is a constantc which depends neither onK, nor on theindicesj1, . . . , ji, nor on the special choice of the orthonormal vectorsu1, . . . , ud−1 ⊥ u. This follows from the invariance ofνu with respectto O(u⊥). By evaluating relation (18) for the unit ball, we conclude thatc = 1/
(d−1
i
), and this yields the statement of the lemma forj = i + 1.
The general case now follows by applying an integral-geometric identitywhich is essentially equivalent to Satz 6.1.1 in [44] and by using twice thespecial case which has been established in the first part of the proof.ut
The following theorem plays a central role in the context of character-izations of absolute continuity for surface area measures. There is also ananalogous result involving the additional assumption of bounded densities,but a precise description and a proof of this statement will be postponed to[25].
Theorem 4.4. Let K ∈ Kd, i ∈ {1, . . . , d − 2}, j ∈ {i + 1, . . . , d − 1},and letω ∈ B(Sd−1). Then
Si(K, ·) x ω � S0(K, ·) x ω
if and only if
SVi (K|V, ·) x (ω ∩ V ) � SV
0 (K|V, ·) x (ω ∩ V ) ,
for νj almost all linear subspacesV ∈ G(d, j).
Proof. First, let us assume that
SVi (K|V, ·) x (ω ∩ V ) � SV
0 (K|V, ·) x (ω ∩ V ) ,
for νj almost allV ∈ G(d, j). But then, forνj almost allV ∈ G(d, j), theequation
SVi (K|V, α ∩ V ) =
∫α∩V
DVi h(K|V, u) Hj−1(du) (19)
holds for any Borel setα ⊆ ω.On the other hand, it is known that the projection formula
Si(K, α) =ωd
ωj
∫G(d, j)
SVi (K|V, α ∩ V ) νj(dV ) (20)
holds for allα ∈ B(Sd−1); see relation (4.5.26) in [41]. Inserting Eq. (19)into Eq. (20), we obtain from Lemma 4.1 and Lemma 4.3 that
Si(K, α) =ωd
ωj
∫G(d, j)
∫Sd−1∩V
1α(u)DVi h(K|V, u) Hj−1(du) νj(dV )
=∫
α
∫Gu(d, j)
DVi h(K|V, u) νu
j (dV ) Hd−1(du)
=∫
αDih(K, u) Hd−1(du) ,
464 D. Hug
whereα ⊆ ω is an arbitrary Borel set. This shows thatSi(K, ·) x ω �S0(K, ·) x ω.
Conversely, assume now thatSi(K, ·) x ω � S0(K, ·) x ω. Employingsuccessively Lemma 4.1, Lemma 4.3, Eq. (2.8) from [24], Eq. (20), and theLebesgue decomposition theorem forSV
i (K|V, ·), we obtain that∫G(d, j)
∫Sd−1∩V
1ω(u)DVi h(K|V, u) Hj−1(u) νj(dV )
=ωj
ωd
∫ω
∫Gu(d, j)
DVi h(K|V, u) νu
j (dV ) Hd−1(du)
=ωj
ωd
∫ω
Dih(K, u) Hd−1(du)
=ωj
ωdSa
i (K, ω) =ωj
ωdSi(K, ω)
=∫G(d, j)
SVi (K|V, ω ∩ V ) νj(dV )
=∫G(d, j)
∫Sd−1∩V
1ω(u)DVi h(K|V, u) Hj−1(du) νj(dV )
+∫G(d, j)
(SV
i
)s(K|V, ω ∩ V ) νj(dV ) .
This yields ∫G(d, j)
(SV
i
)s(K|V, ω ∩ V ) νj(dV ) = 0 .
Hence, forνj almost allV ∈ G(d, j), we obtain(SV
i
)s(K|V, ω ∩ V ) = 0 ,
that is,
SVi (K|V, ·) x (ω ∩ V ) � SV
0 (K|V, ·) x (ω ∩ V ) ,
and this completes the proof. utAs an immediate consequence we obtain:
Theorem 4.5. LetK ∈ Kd, ω ∈ B(Sd−1), andi ∈ {1, . . . , d − 2}. Then
Si(K, ·) x ω � S0(K, ·) x ω
if and only if, for νi+1 almost allU ∈ G(d, i + 1), the projectionK|Uis supported from outside by an(i + 1)-dimensional ball atHi almost allpoints of the setτ(K|U, ω ∩ U).
Curvature measures of convex sets II 465
Proof. This immediately follows from Theorem 3.7 and from a special caseof Theorem 4.4. ut
In the remaining part of this section, we establish a characterization forthe absolute continuity of surface area measures which involves touchingplanes and supporting orthogonal spherical cylinders. This also leads to aregularity result. To achieve this aim we introduce some terminology.
For a convex bodyK ∈ Kd and somer ∈ {0, . . . , d − 1}, the setA(K, d, r) of r-dimensional affine subspaces ofR
d which touchK hasbeen defined in Sect. 2. A parametrization of this rectifiable set is providedin [54] and [35]. We say thatK is supported from outside by an orthogonalspherical cylinder atE ∈ A(K, d, r) if there is someR > 0 and someu ∈ Sd−1 with E ⊆ H(K, u) such thatK ⊆ E + B(−Ru, R).
Weil [51] defines a natural measure onA(K, d, r) in the following way.Let B ∈ B(A(d, r)) andU ∈ G(d, d − r). Then
T (B, U) :={
x ∈ U : x + U⊥ ∈ B}
is a Borel set, and we can define the measure
µr(K, B) :=∫G(d, d−r)
CUd−1−r(K|U, T (B, U)) νd−r(dU) . (21)
The measurability of the integrand was proved by Weil [51]; see also§5.3 in[44]. Althoughµr(K, ·) is defined onB(A(d, r)), the measure is concen-trated on the subsetA(K, d, r). Henceforth, we shall replace the measurespaces(B(A(d, r)), µr(K, ·)) and(G(d, r), νr) by their completions with-out changing our notation. The members of the extendedσ-algebras will becalledµr(K, ·) andνr measurable sets, respectively. It was shown in [51],for K ∈ Kd, r ∈ {0, . . . , d − 1} andω ∈ B(Sd−1), that
τr(K, ω) := {E ∈ A(K, d, r) : E ⊆ H(K, u) for someu ∈ ω}is µr(K, ·) measurable and
Sd−1−r(K, ω) =ωd
ωd−rµr(K, τr(K, ω)) . (22)
The setτr(K, ω) will be called thereverse spherical image of orderr of Kat ω. Thus the reverse spherical image of orderr = 0 is just the ordinaryreverse spherical image.
Equation (22) has previously been used as an integral-geometric inter-pretation for the intermediate surface area measures. In the present context,it shows that it is natural to state a characterization of absolute continuityfor surface area measures by using touching planes.
466 D. Hug
Theorem 4.6. LetK ∈ Kd, ω ∈ B(Sd−1), andi ∈ {1, . . . , d − 2}. Then
Si(K, ·) x ω � S0(K, ·) x ω
if and only ifK is supported from outside by an orthogonal spherical cylinderat µd−1−i(K, ·) almost allE ∈ τd−1−i(K, ω).
Remark 5.In contrast to the two-step procedure of Theorem 4.5, Theorem4.6 provides a one-step procedure for verifying the absolute continuity ofsurface area measures of convex bodies. With regard to Eq. (22), this char-acterization connects the measure theoretic and geometric aspects of theproblem in a natural way. Furthermore, note that the equivalence of condi-tions (a) and (c) of Theorem 3.7 can be viewed as the statement of Theorem4.6 in the casei = d − 1 if properly interpreted.
Proof of Theorem 4.6.The setBd−1−i(K, ω) of all E ∈ τd−1−i(K, ω)such thatK is supported from outside by an orthogonal spherical cylinderat E is µd−1−i(K, ·) measurable. In fact, this set is equal to the set of allE ∈ τd−1−i(K, ω) for which there is somen ∈ N and someu ∈ Sd−1 suchthat
E ⊆ H(K, u) and K ⊆ E + B(−nu, n) . (23)
Therefore it remains to prove that, for eachn ∈ N, the set of all touchingaffine subspacesE ∈ A(K, d, d − 1 − i) for which there is someu ∈ Sd−1
such that condition (23) is satisfied, is closed inA(d, d − 1 − i). But thiscan easily be checked.
Now, let us denote by
Bcd−1−i(K, ω) := τd−1−i(K, ω) \ Bd−1−i(K, ω)
the set of allE ∈ τd−1−i(K, ω) such thatK is not supported from outsideby an orthogonal spherical cylinder atE. From relation (5.2) in Weil [51, p.97] it can be inferred that
µd−1−i(K, Bcd−1−i(K, ω)) = 0
if and only if
Hi(bdU (K|U) ∩ Td−1−i
(Bc
d−1−i(K, ω), U))
= 0 ,
Curvature measures of convex sets II 467
for νi+1 almost allU ∈ G(d, i + 1). Moreover, we can write
bdU (K|U) ∩ Td−1−i
(Bc
d−1−i(K, ω), U)
={
x ∈ bdU (K|U) : x + U⊥ ∈ Bcd−1−i(K, ω)
}={
x ∈ bdU (K|U) :(x + U⊥ ⊆ H(K, v) for somev ∈ ω
)and(
x + U⊥ ⊆ H(K, u) ⇒K 6⊆ x + U⊥ + B(−Ru, R)
)holds for allR > 0 and allu ∈ Sd−1
}={
x ∈ τ(K|U, ω ∩ U) : K|U is not supported from outside
by an(i + 1)-dimensional ball atx}
.
An application of Theorem 4.5 then completes the proof. utThe next theorem demonstrates that the rectifiability of some surface area
measure of a convex bodyK leads to a certain degree of strict convexityfor K. Another precise statement in this direction was established in [24,Theorem 4.8]. Recall that a support planeH(K, u), u ∈ Sd−1, of a convexbodyK is said to be regular ifu is a regular normal vector ofK.
Theorem 4.7. LetK ∈ Kd,ω ∈ B(Sd−1), i ∈ {1, . . . , d−2}, and assumethat
Si(K, ·) x ω � S0(K, ·) x ω .
Then, forµd−1−i(K, ·) almost allE ∈ τd−1−i(K, ω), every support planeof K which containsE is regular.
Proof. Denote byE1 the set of allE ∈ τd−1−i(K, ω) for which
card(E ∩ K) > 1 or card{
u ∈ Sd−1 : E ⊆ H(K, u)}
> 1 .
By a result of Zalgaller [56] (see also Schneider [41,§2.3]), the proof ofwhich is based on methods of Ewald, Larman & Rogers [15], and usingLemma 5.5 of Weil [51], we deduce thatE1 hasµd−1−i(K, ·) measure zero.Further, letE2 be the set of allE ∈ τd−1−i(K, ω)such thatK is not supportedfrom outside by an orthogonal spherical cylinder atE. Theorem 4.6 impliesthatE2 hasµd−1−i(K, ·) measure zero as well.
Now choose anyE ∈ τd−1−i(K, ω) \ (E1 ∪ E2). Then K ⊆ E +B(−Ru, R) holds for someR > 0 and for a uniquely determined vectoru ∈ ω with E ⊆ H(K, u). Therefore,
In this final section, our first aim is to deduce Theorems 2.5 and 2.4 froma sequence of auxiliary results. Then we prove Theorems 2.6 and 2.7. Es-sentially, the basic approach for curvature measures is dual to the one forsurface area measures. Instead of projections onto linear subspaces, whichhave been essential for surface area measures in Sect. 4, we now considerintersections of convex bodies with affine subspaces. Moreover, principalradii of curvature (as functions which are defined almost everywhere on theunit sphere) are replaced by principal curvatures which are defined (almosteverywhere) on the boundary of a given convex body.
However, for curvature measures the situation is more complicated. Forexample, Lemma 5.4 below cannot be obtained by using invariance proper-ties of suitably defined Haar measures, at least not in an obvious way. Thisis in contrast to the proof of Lemma 4.1. Instead one uses Federer’s coareaformula and the alternating calculus of multilinear algebra to establish therequired integral-geometric transformation. A similar remark applies to theproof of Proposition 5.11, for which no analogue is required in Sect. 4. It is aspecial feature of the present work that both results about Haar measures andbasic arguments from geometric measure theory are combined. A secondcomplication arises, since it is not sufficient to consider affine subspaceswhich intersect the boundary of a given convex body orthogonally at a pre-scribed boundary point. As a consequence, even for a smooth convex bodyK ∈ Kd
o (of classC2) the principal curvatures of the intersectionsK ∩E ofK with affine subspacesE passing through a fixed boundary pointx ∈ bdKare not uniformly bounded. In fact, these curvatures approach infinity (pro-vided they are not zero) as the section plane approaches a tangential position.For smooth convex bodies this is implied by Meusnier’s theorem. Lemma5.2 extends this classical result in the present setting.
We introduce some additional notation. LetGd be the motion group ofR
d. Denote byA(d, k), for k ∈ {1, . . . , d−1}, the homogeneousGd-spaceof k-dimensional affine subspaces ofR
d, and letµk be the correspondingHaar measure which is normalized as in Schneider [41]. Also from [41, pp.230–231] we adopt the number [L, L′] in the special case whereL = e⊥,e ∈ Sd−1, L′ = U ∈ G(d, s), s ∈ {2, . . . , d − 1}, andlin{L, L′} = R
d.In this situation one has [e⊥, U ] = |〈e, u〉| if u ∈ Sd−1 ∩ U ∩ V ⊥ andV := e⊥ ∩ U ∈ G(d, s − 1). In particular, the subspacee⊥ will be the(d−1)-dimensional linear tangent spaceTxK of a convex setK at a regularboundary pointx.
By M(K) we denote the set of allnormal boundary pointsof K ∈ Cd0.
The definition of a normal boundary point in Schneider [41],§2.5, involvesthe notion of convergence in the sense of Hausdorff closed limits; see also[38]. This concept is, for example, described in§§1.1–1.4 of Matheron’s
Curvature measures of convex sets II 469
book [31] or in Hausdorff’s classical treatise [20]. Lemma 5.1 below, whichis used for the proof of Lemma 5.2, provides equivalent conditions in thepresent special situation for convergence in the sense of Hausdorff closedlimits.
Lemma 5.1. LetMi, i ∈ N, andM be non-empty closed convex subsets ofR
n, n ≥ 1, with o ∈ Mi for all i ∈ N. Then the following conditions areequivalent fori → ∞:
(a) Mi → M in the sense of Hausdorff closed limits;(b) Mi ∩ B(o, ρ) → M ∩ B(o, ρ) in the sense of Hausdorff closed limits
for all ρ > 0;(c) Mi ∩ B(o, ρ) → M ∩ B(o, ρ) with respect to the Hausdorff metric for
all ρ > 0.
Proof. (a) ⇔ (b) immediately follows, for example, from the definitionsand from Proposition 1-2-3 in Matheron [31]. Note that for the proof of(a) ⇒ (b) one uses the fact thatMi is star-shaped with respect too forall i ∈ N. Further,(b) ⇔ (c) is a consequence of Proposition 1-4-1 andProposition 1-4-4 in [31]. ut
In the following, we shall occasionally attach a prime ‘′ ’ to certainquantities in order to indicate that they have to be calculated with respectto an affine subspace. For example, the quantityH ′
r−1(K ∩ (x + U), x)in Lemma 5.2 is the normalized elementary symmetric function of orderr − 1 of the principal curvatures of the convex bodyK ∩ (x + U) atx withrespect to thes-dimensional affine subspacex + U . See Lemma 5.2 for theprecise assumptions. This lemma represents a generalization of Meusnier’stheorem from classical differential geometry in the non-smooth setting ofconvex geometry; compare Spivak [47, vol. III, p. 276 (7′)].
Lemma 5.2. Let K ∈ Cdo, r ∈ {2, . . . , d − 1}, ands ∈ {r, . . . , d − 1}.
Furthermore, assume thatx ∈ M(K) andU ∈ G(d, s) satisfyU 6⊆ TxK.Thenx ∈ M′(K∩(x+U)). Moreover, ifU0 := lin{σK(x), U∩TxK}, thenthe principal curvatures of the intersectionsK ∩ (x+U) andK ∩ (x+U0)at x are related by
k′i(K ∩ (x + U), x) = [TxK, U ]−1k′
i(K ∩ (x + U0), x) ,
for i ∈ {1, . . . , s − 1}, and they correspond to the same directions of thecommon tangent spaceTxK ∩ U . In particular,
H ′r−1(K ∩ (x + U), x) = [TxK, U ]1−rH ′
r−1(K ∩ (x + U0), x) .
Proof. All limits in the proof are meant in the sense of Hausdorff closedlimits. We can assume thatx = o. Let ed := −σK(x) andV := U ∩ TxK.
470 D. Hug
Further, chooseλ > 0, es(λ) ∈ U ∩ V ⊥ ∩ Sd−1 andes ∈ Sd−1 ∩ e⊥d ∩ V ⊥
such that
U = lin{V, es(λ)} and es(λ) =ed + λes√
1 + λ2.
We setUλ := U and define
S(h) := K ∩(e⊥d + hed
)− hed .
Sincex ∈ M(K), there exists a closed setD ⊆ e⊥d such that
limh↓0
1√2h
S(h) = D , (24)
and the boundary (if any) ofD is a quadric.Now set
Sλ(h) := K ∩ Uλ ∩ (V + hes(λ)) − hes(λ) .
From Eq. (24), Lemma 5.1 and Theorem 1.8.8 in Schneider [41] we concludethat
limh↓0
1√2h
S0(h) = D ∩ V ,
and the boundary (if any) ofD ∩ V is a quadric.Observe that [TxK, Uλ] = 〈ed, es(λ)〉 and
Sλ
(〈ed, es(λ)〉−1h)
=[K ∩
(e⊥d + hed
)− hed
]∩ [V + λhes] − λhes .
Hence,
1√2h
Sλ
(〈ed, es(λ)〉−1h)
=[
1√2h
(K ∩
(e⊥d + hed
)− hed
)]∩(
V + λ
√h
2es
)− λ
√h
2es .
Again Eq. (24), Lemma 5.1 and Theorem 1.8.8 in [41] imply that
limh↓0
1√2h
Sλ
(〈ed, es(λ)〉−1h)
= limh↓0
1√2h
S0(h) = D ∩ V .
Thus, we obtain that
limh↓0
1√2h〈ed, es(λ)〉−1
Sλ
(〈ed, es(λ)〉−1h)
=√
[TxK, U ](D ∩ V ) ,
and the boundary (if any) of√
[TxK, U ](D ∩ V ) is a quadric. This yieldsthe statement of the lemma. ut
Curvature measures of convex sets II 471
The next two lemmas will be needed to justify the application of Fubini’stheorem and to perform certain integral-geometric transformations in thecourse of the proofs of Proposition 5.8 and Theorem 2.5.
Lemma 5.3. Let K ∈ Cdo, r ∈ {2, . . . , d − 1}, ands ∈ {r, . . . , d − 1}.
Then the following statements hold:
(1) D2 := {(x, U) ∈ bdK ×G(d, s) : x ∈ M(K), (x+U)∩ intK 6= ∅}is a Borel set;
(2) (x, U) 7→ H ′r−1(K ∩ (x + U), x) is Borel measurable onD2;
(3) νs({U ∈ G(d, s) : (x + U) ∩ intK = ∅}) = 0 if x ∈ reg K.
Proof. The proof follows from standard methods of measure theory andconvex geometry; compare also the proof of Lemma 4.2. For the proof ofthe second statement one can use Lemma 5.2. utLemma 5.4. LetK ∈ Cd
o, s ∈ {2, . . . , d − 1}, andf : bdK × G(d, s) →[0,∞] be Borel measurable. Then∫
bd K
∫G(d, s)
[TxK, U ]f(x, U) νs(dU) Hd−1(dx)
=∫A(d, s)
∫bd K∩E
f(x, U(E)) Hs−1(dx) µs(dE) ,
whereU(E) ∈ G(d, s) is the unique linear subspace which is parallel toE.
Proof. This is a special case of Theorem 1 in Zahle [55]. Observe thatµs
almost alls-dimensional affine subspacesE ∈ A(d, s) which meetK alsomeetintK. ut
The following three lemmas, which will be essential for the proof ofProposition 5.8, are based on integral-geometric transformations. In orderto state and prove these lemmas, we introduce some further definitions.
Let s ∈ {2, . . . , d − 1} andW ∈ G(d, d − 1). Then we set
G(W, s − 1) := {V ∈ G(d, s − 1) : V ⊆ W}and denote byνW
s−1 the corresponding normalized Haar measure ofG(W, s−1) which is invariant with respect toO(W ). Moreover, ifj ∈ {s, . . . , d−1}andV ∈ G(d, s − 1), then
GV (d, j) := {U ∈ G(d, j) : V ⊆ U} ,
andνVj is the corresponding normalized Haar measure ofGV (d, j) which
is invariant with respect to all rotationsρ ∈ O(d) for whichρ(v) = v holdsfor all v ∈ V .
472 D. Hug
Lemma 5.5. Let K ∈ Cdo, r ∈ {2, . . . , d − 1}, s ∈ {r, . . . , d − 1}, and
assume thatx ∈ M(K). In addition, sete := σK(x) ∈ Sd−1. Then
Hr−1(K, x) =∫G(e⊥, s−1)
H ′r−1(K ∩ (x + lin{e, V }), x)
×νe⊥s−1(dV ) . (25)
Proof. The proof is essentially the same as the one for Lemma 4.3.utLemma 5.6. Let e ∈ Sd−1, s ∈ {2, . . . , d − 1}, and leth : G(d, s) →[0,∞] be Borel measurable. Then
2 ωd
ωs ωd−s+1
∫G(d, s)
h(U) νs(dU)
=∫G(e⊥, s−1)
∫GV (d, s)
[e⊥, U ]s−1h(U) νVs (dU) νe⊥
s−1(dV ) .
Proof. The proof will be accomplished by applying Satz 6.1.9 from Schnei-der & Weil [44]. Leth : G(d, s) → [0,∞) andg : G(d, d−1) → [0,∞) bearbitrary continuous functions. Furthermore, setf(U, W ) := h(U)g(W ),for anyU ∈ G(d, s) andW ∈ G(d, d − 1).
In the following, we shall repeatedly apply Fubini’s theorem. The re-quired measurability can be established in the same way as in the proof ofHilfssatz 7.2.4 of [44]. Then Satz 6.1.9 and Satz 6.1.1 from [44] imply that
(cd s (d−1)
)−1∫G(d, s)
h(U) νs(dU)∫G(d, d−1)
g(W ) νd−1(dW )
=∫G(d, s−1)
∫GV (d, s)
∫GV (d, d−1)
[U, W ]s−1h(U)g(W )
× νVd−1(dW ) νV
s (dU) νs−1(dV )
=∫G(d, d−1)
∫G(W, s−1)
∫GV (d, s)
[U, W ]s−1h(U)g(W )
× νVs (dU) νW
s−1(dV ) νd−1(dW )
=∫G(d, d−1)
g(W )H(W ) νd−1(dW ) .
The functionH : G(d, d − 1) → [0,∞) is defined by
H(W ) :=∫G(W, s−1)
∫GV (d, s)
[U, W ]s−1h(U) νVs (dU) νW
s−1(dV ) .
It can be shown thatH is continuous. This follows by applying twice an ar-gument which is similar to the one used to verify Hilfssatz 7.2.4 in Schneider
Curvature measures of convex sets II 473
& Weil [44]. In fact, one defines
G(d, d − 1, s − 1) := {(V, W ) ∈ G(d, s − 1) × G(d, d − 1) : V ⊆ W}and starts by proving that the map
G(d, d − 1, s − 1) → [0,∞),
(V, W ) 7→∫GV (d, s)
[U, W ]s−1h(U) νVs (dU) ,
is continuous.Sinceg was arbitrarily chosen andH is continuous, we thus conclude
that the relation∫G(d, s)
h(U) νs(dU) = cd s (d−1)H(W )
holds for an arbitraryW ∈ G(d, d − 1). ChoosingW := e⊥ and notingthat
cd s (d−1) =ωd−s+1 ωs
ωd ω1,
we obtain the statement of the lemma for a continuous functionh. But thenthe general result follows by standard approximation arguments. utRemark 6.Lemma 5.6 can also be proved by applying the coarea formulato the map
T : G(d, s)∗ → G(e⊥, s − 1) , U 7→ e⊥ ∩ U ,
where
G(d, s)∗ := {U ∈ G(d, s) : U 6⊆ e⊥} .
For this approach one has to check thatT is differentiable and that
J(s−1)(d−s)T (U) = [e⊥, U ]1−s
for all U ∈ G(d, s)∗.
In Lemma 5.7 and subsequently we writeκn for the volume of then-dimensional unit ball,n ≥ 0, that is,κn = πn/2/Γ (1 + n/2).
Lemma 5.7. Let e ∈ Sd−1, r ∈ {2, . . . , d − 1}, s ∈ {r, . . . , d − 1}, andchoose someV ∈ G(e⊥, s − 1). Then∫
GV (d, s)[e⊥, U ]s−r+1 νV
s (dU) =2
ωd−s+1
κd−r
κs−r.
474 D. Hug
Proof. Let the assumptions of the lemma be fulfilled. Then, using the intro-ductory remarks of Chap. 6 in Schneider & Weil [44], we obtain∫
GV (d, s)[e⊥, U ]s−r+1 νV
s (dU)
=∫G(V ⊥, 1)
[e⊥, lin{V, L}]s−r+1 νV ⊥1 (dL)
=∫
Sd−1∩V ⊥|〈e, u〉|s−r+1(ωd−s+1)−1 Hd−s(du)
=1
ωd−s+1
∫Sd−1−s
∫ π
0| cos ϕ|s−r+1 | sinϕ|d−1−s dϕ Hd−1−s(du)
=ωd−s
ωd−s+12∫ π/2
0(cos ϕ)s−r+1(sinϕ)d−1−s dϕ
=2
ωd−s+1
κd−r
κs−r,
and this completes the proof. utThe following proposition represents the main tool for establishing The-
orem 2.5.
Proposition 5.8. Let K ∈ Cdo, r ∈ {2, . . . , d − 1}, s ∈ {r, . . . , d − 1},
and assume thatx ∈ M(K). Then
Hr−1(K, x) = adsr
∫G(d, s)
[TxK, U ]
×H ′r−1(K ∩ (x + U), x) νs(dU) , (26)
where
adsr :=κs−r ωd
κd−r ωs.
Proof. Let e ∈ (TxK)⊥ ∩ Sd−1. By successively applying Lemma 5.6,Lemma 5.2, and finally Lemma 5.7 as well as Lemma 5.5, we obtain that
2 ωd
ωs ωd−s+1
∫G(d, s)
[TxK, U ]H ′r−1(K ∩ (x + U), x) νs(dU)
=∫G(e⊥, s−1)
∫GV (d, s)
[e⊥, U ]s
×H ′r−1(K ∩ (x + U), x) νV
s (dU) νe⊥s−1(dV )
=∫G(e⊥, s−1)
∫GV (d, s)
[e⊥, U ]s−r+1
Curvature measures of convex sets II 475
×H ′r−1(K ∩ (x + lin{e, V }), x) νV
s (dU) νe⊥s−1(dV )
=∫G(e⊥, s−1)
H ′r−1(K ∩ (x + lin{e, V }), x)
×∫GV (d, s)
[e⊥, U ]s−r+1 νVs (dU) νe⊥
s−1(dV )
=2
ωd−s+1
κd−r
κs−rHr−1(K, x) .
This yields the desired result. utNow we have completed the preparations for the proofs of Theorems 2.5
and 2.4.
Proof of Theorem 2.5.It is sufficient to assume thatK ∈ Kdo, since the
curvature measures are locally defined. Moreover, we shall repeatedly useFubini’s theorem without further mentioning it. The required measurabilityis guaranteed by Lemma 5.3.
First, we assume that forµs almost allE ∈A(d, s)such thatE ∩ intK 6= ∅the relation
C ′s−r(K ∩ E, ·) x (β ∩ E) � C ′
s−1(K ∩ E, ·) x (β ∩ E)
is satisfied. Letγ ⊆ β be an arbitrary Borel set. Then we obtain fromthe Crofton intersection formula, Theorem 4.5.5 in Schneider [41, p. 235],from the assumption and Eq. (2.7) of [24] applied ins-dimensional affinesubspacesE, and from Lemma 5.4 that
Cd−r(K, γ)
= adsr
∫A(d, s)
C ′s−r(K ∩ E, γ ∩ E) µs(dE)
= adsr
∫A(d, s)
∫bd K∩E
1γ(x)H ′r−1(K ∩ E, x) Hs−1(dx) µs(dE)
= adsr
∫bd K
1γ(x)∫G(d, s)
[TxK, U ]
×H ′r−1(K ∩ (x + U), x) νs(dU) Hd−1(dx)
=∫
bd K∩γHr−1(K, x) Hd−1(dx) .
Note that the last equation is implied by Proposition 5.8. Thus
Cd−r(K, ·) x β � Cd−1(K, ·) x β ,
sinceγ was an arbitrary Borel subset ofβ.
476 D. Hug
Now we assume thatCd−r(K, ·) x β � Cd−1(K, ·) x β. Using Lemma5.4, Proposition 5.8, Eq. (2.7) from [24], the assumption of the theorem, The-orem 4.5.5 from Schneider [41], and the Lebesgue decomposition theoremapplied toC ′
s−r(K ∩ E, ·), we obtain that
∫A(d, s)
∫bd K∩E
1β(x)H ′r−1(K ∩ E, x) Hs−1(dx) µs(dE)
=∫
bd K
∫G(d, s)
1β(x)[TxK, U ]
×H ′r−1(K ∩ (x + U), x) νs(dU) Hd−1(dx)
=1
adsr
∫bd K
1β(x)Hr−1(K, x) Hd−1(dx)
=1
adsrCa
d−r(K, β) =1
adsrCd−r(K, β)
=∫A(d, s)
C ′s−r(K ∩ E, β ∩ E) µs(dE)
=∫A(d, s)
∫bd K∩E
1β(x)H ′r−1(K ∩ E, x) Hs−1(dx) µs(dE)
+∫A(d, s)
(C ′s−r)
s(K ∩ E, β ∩ E) µs(dE) .
Hence, forµs almost allE ∈ A(d, s) such thatintK ∩E 6= ∅, the singularpart of the measureC ′
s−r(K ∩E, ·) x (β ∩E) vanishes. This establishes theconverse part of the theorem. utProof of Theorem 2.4.Forr = d the theorem has already been verified. Thuswe can assume thatr ∈ {2, . . . , d−1}. But then the statement follows fromTheorem 2.3 and a special case of Theorem 2.5. ut
The following three auxiliary results pave the way to the proof of The-orem 2.6. The first of these is of a purely geometric nature, the other twolemmas are integral-geometric results.
Lemma 5.9. Let K ∈ Kdo, r ∈ {2, . . . , d − 1}, E ∈ A(K, d, r − 1), and
let p ∈ E ∩ K. Then the implication
(E + u−) ∩ int K 6= ∅ ⇒(
B(p − ru, r) ∩ (E + u−) ⊆ K for somer > 0)
holds for someu ∈ Sd−1 ∩ U(E)⊥ with (E + u−) ∩ int K 6= ∅ if and onlyif the implication holds for allu ∈ Sd−1 ∩ U(E)⊥.
Curvature measures of convex sets II 477
Proof. It can be assumed thatp = o andE = lin{e1, . . . , er−1}. Let thevectorsui ∈ Sd−1 ∩ U(E)⊥, i ∈ {1, 2}, be linearly independent and suchthat (
E + u−i
) ∩ int K 6= ∅ , for i ∈ {1, 2} .
Furthermore, suppose that
B(−ru1, r) ∩ (E + u−1) ⊆ K
for somer > 0. Let y ∈ (E + u−2) ∩ int K. In particular,y can be chosen
such thaty /∈ E. Then, ifε > 0 is sufficiently small, we obtain that
x := y + ε(y + ru1) ∈ int K .
Hence we have
conv{x, B(−ru1, r) ∩ (E + u−
1)} ∩ (E + u−
2) ⊆ K , (27)
and it is sufficient to show that the set on the left-hand side of (27) is anellipsoid, since a ball of a suitably small radius will roll freely inside anygiven ellipsoid.
In order to prove this assertion, leter ∈ lin{u1, u2, E}∩Sd−1 ∩ lin{u1,E}⊥ be such that〈x, er〉 > 0. Further, letα be a linear map of lin{u1, u2, E}onto itself which leaves lin{u1, E} invariant and which satisfies
α(x) = −ru1 + 〈x, er〉er .
This yields thatα(y) = −ru1 + 〈y, er〉er 6= o. In addition, we know thaty = e − λ0u2 with somee ∈ E and some positive constantλ0. Therefore,
α(conv
{x, B(−ru1, r) ∩ (E + u−
1)} ∩ (E + u−
2))
= conv{α(x), B(−ru1, r) ∩ (E + u−
1)} ∩ (E + (−α(y))−) , (28)
sinceα(E + u−2 ) = E + (−α(y))−. It is a well-known fact of elementary
geometry that the set on the right-hand side of (28) is an ellipsoid. Thus, byapplyingα−1 to Eq. (28) the assertion follows. utLemma 5.10. Letr ∈ {2, . . . , d−1}, and letf : G(d, r−1)×Sd−1 → R
be a non-negative Borel measurable function. Then∫G(d, r−1)
∫Sd−1∩U⊥
f(U, u) Hd−r(du) νr−1(dU)
=ωd−r+1
ωr
∫G(d, r)
∫Sd−1∩V
f(V ∩ u⊥, u) Hr−1(du) νr(dV ) .
478 D. Hug
Proof. The set
G∗ :={
(U, u) ∈ G(d, r − 1) × Sd−1 : u ∈ U⊥}
together with the operation
O(d) × G∗ → G∗ , (ρ, (U, u)) 7→ (ρU, ρu) ,
is a homogeneousO(d)-space. Using the fact that the map
{(V, u) ∈ G(d, r) × Sd−1 : u ∈ V } → G∗ , (V, u) 7→ (V ∩ u⊥, u) ,
is Borel measurable, we can define two measures onG∗ by setting
µ1(A) :=∫G(d, r−1)
∫Sd−1∩U⊥
1A(U, u) Hd−r(du) νr−1(dU)
and
µ2(A) :=∫G(d, r)
∫Sd−1∩V
1A(V ∩ u⊥, u) Hr−1(du) νr(dV )
for A ∈ B(G∗). These two measures areO(d)-invariant. In fact, for anyθ ∈ O(d) we deduce from theO(d)-invariance ofνr−1 andHd−r that
µ1(A) =∫G(d, r−1)
∫Sd−1∩U⊥
1A(U, u) Hd−r(du) νr−1(dU)
=∫G(d, r−1)
∫Sd−1∩(θU)⊥
1A(θU, u) Hd−r(du) νr−1(dU)
=∫G(d, r−1)
∫Sd−1∩U⊥
1A(θU, θu) Hd−r(du) νr−1(dU)
= µ1(θ−1A) ,
and a similar argument can be given forµ2. Hence, by the uniqueness theo-rem for Haar measures, we conclude thatµ1 = c µ2 with a positive constantc. The explicit value ofc follows from substitutingA = G∗. ut
For the statement of the following proposition, which plays a crucial rolein the proof of Theorem 2.6, two further definitions will be needed.
Let K ∈ Kdo andr ∈ {2, . . . , d − 1}. Then we set
AK(d, r)∗ := {F ∈ A(d, r) : F ∩ int K 6= ∅}and
A(K, d, r − 1, 1)∗ :={
(E, u) ∈ A(K, d, r − 1) × Sd−1 :
u ∈ U(E)⊥, (E + u−) ∩ int K 6= ∅}
.
Curvature measures of convex sets II 479
In a certain sense, the next result, Proposition 5.11, provides a tool fortranslating statements about(r − 1)-dimensional touching affine subspacesinto statements aboutr-dimensional intersecting affine subspaces, and viceversa.
Proposition 5.11. Let K ∈ Kdo, r ∈ {2, . . . , d − 1}, and assume that the
functionf : A(K, d, r − 1) × Sd−1 → [0,∞] is Borel measurable. Then
∫A(K, d, r−1)
∫Sd−1∩U(E)⊥
1A(K, d, r−1, 1)∗(E, u)f(E, u)
×Hd−r(du) µr−1(K, dE)
=ωd−r+1
ωr
∫AK(d, r)∗
∫Sd−1∩U(F )
J(F, u)f(HF (K ∩ F, u), u)
×Hr−1(du) µr(dF ) ,
where
J(F, u) :=⟨σK|lin{u,U(F )⊥}
(HF (K ∩ F, u) ∩ lin
{u, U(F )⊥
}), u⟩−1
is well-defined forµr almost allF ∈ AK(d, r)∗ andHr−1 almost all unitvectorsu ∈ Sd−1 ∩ U(F ).
Proof. The proof is accomplished by a sequence of integral-geometric trans-formations. First, note thatAK(d, r)∗ andA(K, d, r−1, 1)∗ can be writtenas countable unions of closed sets. This follows from choosingKn ∈ Kd,for n ∈ N, with Kn ⊂ Kn+1 and intK = ∪n≥1Kn. Then, using the defini-tion of µr−1(K, ·), the fact thatνd−r+1 is the image measure ofνr−1 underthe mapG(d, r − 1) → G(d, d − r + 1), U 7→ U⊥, Fubini’s theorem, and
480 D. Hug
Lemma 5.10, we obtain∫A(K, d, r−1)
∫Sd−1∩U(E)⊥
1A(K, d, r−1, 1)∗(E, u)f(E, u)
×Hd−r(du) µr−1(K, dE)
=∫G(d, d−r+1)
∫bdU (K|U)
∫Sd−1∩U
1A(K, d, r−1, 1)∗(x + U⊥, u)
×f(x + U⊥, u)Hd−r(du) Hd−r(dx) νd−r+1(dU)
=∫G(d, r−1)
∫bd
U⊥ (K|U⊥)
∫Sd−1∩U⊥
1A(K, d, r−1, 1)∗(x + U, u)
×f(x + U, u)Hd−r(du) Hd−r(dx) νr−1(dU)
=∫G(d, r−1)
∫Sd−1∩U⊥
∫bd
U⊥ (K|U⊥)1A(K, d, r−1, 1)∗(x + U, u)
×f(x + U, u)Hd−r(dx) Hd−r(du) νr−1(dU)
=ωd−r+1
ωr
∫G(d, r)
∫Sd−1∩V
∫bdlin{u,V ⊥}(K|lin{u,V ⊥})
×1A(K, d, r−1, 1)∗(x + V ∩ u⊥, u)f(x + V ∩ u⊥, u)
× Hd−r(dx) Hr−1(du) νr(dV ) .
For any fixedV ∈ G(d, r) andu ∈ Sd−1 ∩ V , we define the map
GV,uK : bdlin{u,V ⊥}
(K|lin
{u, V ⊥
})→ K|V ⊥ , y 7→ y − 〈y, u〉u .
The restriction ofGV,uK to the set of allx ∈ bdlin{u,V ⊥}
(K|lin {u, V ⊥})
such that(x + (V ∩ u⊥) + u−) ∩ int K 6= ∅ is a bijection onto intV ⊥
(K
|V ⊥). Moreover, forHd−r almost allx ∈ bdlin{u,V ⊥}(K|lin {u, V ⊥})
such that(x + (V ∩ u⊥) + u−) ∩ int K 6= ∅, one obtains that
Jd−r GV,uK (x) =
∣∣∣∣⟨
σlin{u,V ⊥}K|lin{u,V ⊥}(x), u
⟩∣∣∣∣ > 0 .
Therefore an application of the coarea formula shows that the precedingchain of equalities can be continued with
ωd−r+1
ωr
∫G(d, r)
∫Sd−1∩V
∫int
V ⊥(K|V ⊥)
{Jd−r GV,u
K
((GV,u
K
)−1(y))}−1
×f
((GV,u
K
)−1(y) + V ∩ u⊥, u
)Hd−r(dy) Hd−r(du) νr(dU) .
Curvature measures of convex sets II 481
But (GV,u
K
)−1(y) + V ∩ u⊥ = H(y+V )(K ∩ (y + V ), u) ,
and forHd−r almost ally ∈ intV ⊥(K|V ⊥) the(d − r)-dimensional Jaco-
bian
Jd−r GV,uK
((GV,u
K
)−1(y))
is given by ∣∣∣∣⟨
σlin{u,U(y+V )⊥}K|lin{u,U(y+V )⊥}
(H(y+V )(K ∩ (y + V ), u)
∩(
lin{
u, U(y + V )⊥}))
, u
⟩∣∣∣∣−1
.
It is convenient to write the argument of the spherical image map as a setwhich consists of precisely one point. This slight abuse of notation shouldnot lead to any misunderstanding. Hence, the proof is completed by usingonce again Fubini’s theorem and the representation ofµr given in§4.5 of[41]. utProof of Theorem 2.6.Let K, β, andr be chosen as in the assumptions ofTheorem 2.6. In [51], it was shown that there are setsA1, A2 ∈ B(A(d, r))with µr−1(K, A2) = 0 such that
A1 ⊆ σr−1(K, β) ⊆ A1 ∪ A2 .
By Acr−1(K, β) we denote the set of allE ∈ σr−1(K, β) such thatK is not
supported from inside by anr-dimensional ball atE. Moreover, we writeAc
r−1,1(K, β) for the set of all(E, u) ∈ σr−1(K, β) × Sd−1 such thatu ∈U(E)⊥, (E+u−)∩ int K 6= ∅, and such thatB(p−ρu, ρ)∩(E+u−) 6⊆ Kholds for allp ∈ K ∩ E and allρ > 0. With these definitions we see thatthe inclusions
A1 ∩ Acr−1(K, Rd) ⊆ Ac
r−1(K, β)
⊆(A1 ∩ Ac
r−1(K, Rd))
∪(A2 ∩ Ac
r−1(K, Rd))
,
(A1 × Sd−1) ∩ Acr−1,1(K, Rd) ⊆ Ac
r−1,1(K, β)
and
Acr−1,1(K, β) ⊆
((A1 × Sd−1) ∩ Ac
r−1,1(K, Rd))
∪((A2 × Sd−1) ∩ Ac
r−1,1(K, Rd))
482 D. Hug
are satisfied. Observe thatAcr−1(K, Rd) andAc
r−1,1(K, Rd) are Borel mea-surable sets. Sinceµr−1(K, A2) = 0, we obtain that
µr−1(K, Acr−1(K, β)) = 0
if and only if
µr−1(K, A1 ∩ Acr−1(K, Rd)) = 0 . (29)
Lemma 5.9 yields that Eq. (29) is equivalent to
0 =∫A(K, d, r−1)
∫Sd−1∩U(E)⊥
1(A1×Sd−1)∩Acr−1,1(K, Rd)(E, u)
× Hd−r(du) µr−1(K, dE) . (30)
From Proposition 5.11 and the fact thatJ(F, u) ∈ (0,∞), for µr almost allF ∈ AK(d, r)∗ and forHr−1 almost allu ∈ Sd−1 ∩ U(F ), we obtain thatEq. (30) holds if and only if
0 =∫AK(d, r)∗
∫Sd−1∩U(F )
1(A1×Sd−1)∩Acr−1,1(K, Rd)(H
F (K ∩ F, u), u)
× Hr−1(du) µr(dF ) . (31)
The corresponding integral withA1 replaced byA2 also vanishes, sinceµr−1(K, A2) = 0. Therefore Eq. (31) is equivalent to
Forµr almost allF ∈ AK(d, r)∗
and forHr−1 almost allu ∈ Sd−1 ∩ U(F ),(HF (K ∩ F, u), u
)/∈ Ac
r−1,1(K, β).
(32)
But obviously condition (32) is equivalent to
Forµr almost allF ∈ A(d, r) such thatF ∩ int K 6= ∅and forHr−1 almost allu ∈ σ′(K ∩ F, β ∩ F )
the intersectionK ∩ F is supported from inside
by anr-dimensional ball in directionu.
(33)
Finally, an application of Theorem 2.4 shows that condition (33) is equivalentto
Cd−r(K, ·) x β � Cd−1(K, ·) x β ,
which was to be proved. utProof of Theorem 2.7.Denote byE3 the set of allE ∈ σr−1(K, β) such thatcard(E ∩ K) > 1, let E4 be the set of allE ∈ σr−1(K, β) such thatK is
Curvature measures of convex sets II 483
not supported from inside by anr-dimensional ball atE, and letE5 be theset of allE ∈ σr−1(K, β) such that the pointp which is defined by
{p} = E ∩ bdU(E)⊥(K|U(E)⊥
)is not a regular boundary point ofK|U(E)⊥. Then the result of Zalgaller[56], Theorem 2.6, and Theorem 2.2.4 from Schneider [41] together withthe definition ofµr−1(K, ·) in (21) imply that
µr−1(K, E3 ∪ E4 ∪ E5) = 0 .
Now, chooseE ∈ σr−1(K, β) \ (E3 ∪ E4 ∪ E5), and letx be defined by{x} = E ∩ K. Let S(K, x) denote the support cone ofK at x; see [41,p. 70] for a definition. SinceE /∈ E4, we deduce thatU(E) ⊆ S(K, x),and henceN(K, x) ⊆ U(E)⊥. Furthermore,E /∈ E5 finally implies thatdim N(K, x) = 1, since otherwise the orthogonal projection ofx ontoU(E)⊥ is not a regular boundary point ofK|U(E)⊥. ut
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