PROPERTIES OF DISTANCE FUNCTIONSON CONVEX SURFACES AND ALEXANDROVSPACESJAN RATAJ AND LUD�EK ZAJ�I�CEK

Abstract. If X is a convex surface in a Euclidean space, then the squared(intrinsic) distance function dist2(x; y) is d.c. (DC, delta-convex) on X�Xin the only natural extrinsic sense. For the proof we use semiconcavity (inan intrinsic sense) of dist2(x; y) on X � X if X is an Alexandrov spacewith nonnegative curvature. Applications concerning r-boundaries (dis-tance spheres) and the ambiguous locus (exoskeleton) of a closed subset ofa convex surface are given.

1. IntroductionSemiconcave, and (more general) d.c. functions were recently used in severalworks on the theory of Alexandrov spaces (cf. [12], [17], [18]). The notionof a semiconcave function (or surface) is very old (see, e.g., [8], [20]) andimportant (cf. [5]). A more general notion (see De�nition 2.9 below) of a d.c.function (of many variables) was probably �rst studied by A.D. Alexandrov[1], [2], who studied d.c. surfaces as a natural joint generalization of classicalsmooth surfaces and convex surfaces. In 1959, P. Hartman [10] proved thata locally d.c. function on Rn is d.c. and that d.c. functions (and mappings)are stable with respect to compositions. In the unpublished preprint [17], G.Perelman found independently these results and applied them in the theory ofn-dimensional Alexandrov spaces X with curvature bounded below, namely heshowed that on an open dense subset of X it is possible to introduce naturallya d.c. structure (i.e., the structure of a d.c. manifold), and so to improve someresults of [16]. It seems that the notion of a d.c. manifold has not been studiedin any published work (cf. [12, p. 2]), but some results on d.c. manifolds in Rnwere proved in the unpublished diploma thesis [6] (supervised by the secondauthor).1991 Mathematics Subject Classi�cation. 53C45, 52A20.Key words and phrases. distance function, convex surface, Alexandrov space, d.c. mani-fold, ambiguous locus, skeleton, r-boundary.The research was supported by the grant MSM 0021620839 from the Czech Ministry ofEducation. The second author was also supported by the grant GA�CR 201/06/0198.1

2 J. Rataj and L. Zaj���cekThe main result of the present article (Theorem 4.3) says that if X � Rnis a convex surface, then the squared (intrinsic) distance function g(x; y) =dist2(x; y) is d.c. on X �X in the natural extrinsic sense. It means that g isd.c. onX�X which is equipped by the natural d.c. structure. A weaker versionof this result (in the case n = 3) was known for more then ten years to thesecond author, who used a method similar to that of Alexandrov's proof (fortwo-dimensional convex surfaces) of Alexandrov-Toponogov theorem, namelyan approximation of a general convex surface by polyhedral convex surfacesand considering the devoloping (unfolding) of these polyhedral convex surfaces\along geodesics". However, it is not easy to formalize this geometricallytransparent method (even for n = 3).In the present article we use another method suggested by the �rst author.Namely, we use well-known results and methods from the theory of lengthspaces X with curvature bounded from below (Alexandrov spaces). In theproof, we essentially need Theorem 3.2 on semiconcavity of squared distancefunction dist2(x; y) on X �X (in the intrinsic sense). Although some resultswhich are close to this theorem are known ([15], [19], [24], see Remark 3.3),we were not able to �nd it in the literature, and so we hope that it is of someindependent interest. Using Theorem 3.2, we got rid of using any devolop-ings (unfoldings). However, our proof still needs approximation by polyhedralsurfaces.In Section 5 we present two applications of Proposition 4.2 (which is aquantitative version of our main result Theorem 4.3) concerning r-boundaries(distance spheres) and the ambiguous locus (exoskeleton) of a closed subset ofa convex surface. Note that r-boundaries and ambiguous loci were studied (inEuclidean, Riemannian and Alexandrov spaces) in a number of articles (see,e.g., [9], [22], [30], [11]).In the last short Section 6 we present several remarks and questions con-cerning d.c. structures on length spaces.

2. PreliminariesIn a metric space, B(c; r) denotes the open ball with center c and radius r.If a; b 2 Rn, then [a; b] denotes the segment joining a and b. If F is a Lipschitzmapping, then LipF stands for the least Lipschitz constant of F .If W is a unitary space and V is a subspace of W , then we denote by V ?Wthe orthogonal complement of V in W .If f is a mapping from a normed space X to a normed space Y , then thesymbol df(a) stands for the (Fr�echet) di�erential of f at a 2 X. If df(a) existsand limx;y!a;x 6=y f(y)� f(x)� df(a)(y � x)ky � xk = 0;

Properties of distance functions 3then we say that f is strictly di�erentiable at a (cf. [14, p. 19]).A metric space (X; d) is called a length (or inner or intrinsic) space if, foreach x; y 2 X, d(x; y) equals to the in�mum of lengths of curves joining xand y (see [3, p. 38] or [19, p. 824]). If X is a length space, then a curve' : [a; b] ! X is called minimal, if it is a shortes curve joining its endpointsx = '(a) and y = '(b) parametrized by the arc-length. A length space X iscalled geodesic (or strictly intrinsic) space if each pair of points in X can bejoined by a minimal curve. Note that any complete, locally compact lengthspace is geodesic (see [19, Theorem 8]).Alexandrov spaces with curvature bounded from below are de�ned as lengthspaces which have a lower curvature bound in the sense of Alexandrov. Theprecise de�nition of these spaces can be found in [3] or [19]. (FrequentlyAlexandrov spaces are supposed to be complete and/or �nite dimensional.)If X is a length space and ' : [a; b] ! X a minimal curve, then the points = '((a + b)=2) is called the midpoint of the minimal curve '. A point t iscalled a midpoint of x; y if it is the midpoint of a minimal curve ' joining xand y. If ' as above can be chosen to lie in a set G � X, we will say that t isa G-midpoint of x; y.2.1. Semiconcave functions. For the sake of brevity, we introduce the fol-lowing notation (we use the symbol �2, though �2f(x; y) is one half of asecond di�erence).De�nition 2.1. If f is a real function de�ned on a subset U of a vector spaceand x; y; x+y2 2 U , we denote(1) �2f(x; y) := f(x) + f(y)2 � f �x+ y2

� :Note that, if f(y) = kyk2; y 2 Rn, then

(2) �2f(x+ h; x� h) = kx+ hk2 + kx� hk22 � kxk2 = khk2:Using the above notation, one of several natural equivalent de�nitions (see[5, De�nition 1.1.1 and Proposition 1.1.3]) of semiconcavity in Rn reads asfollows.De�nition 2.2. A function u on an open set A � Rn is called semiconcavewith a semiconcavity constant c � 0 if u is continuous on A and(3) �2u(x+ h; x� h) � (c=2)khk2;whenever x; h 2 Rn and [x� h; x+ h] � A.Remark 2.3. It is well-known and easy to see (cf. [5, Proposition 1.1.3]) thatu is semiconcave on A with semiconcavity constant c if and only if the functiong(x) = u(x)� (c=2)kxk2 is locally concave on A.

4 J. Rataj and L. Zaj���cekWe shall need the following easy lemma. Its �rst part is an obvious conse-quence of [23, Lemma 1.16] (which works with convex functions). The secondpart clearly follows from the �rst one.Lemma 2.4. (i) Let f : (a; b) ! R be a continuous function. Supposethat for every t 2 (a; b) and � > 0 there exists 0 < d < � such that�2f(t+ d; t� d) � 0. Then f is concave on (a; b).(ii) Let f be a continuous function on on open convex subset C � Rn.Suppose that for every x 2 C there exists � > 0 such that �2f(x +h; x� h) � 0 whenever khk < �. Then f is concave on C.As an easy corollary we obtain (via Remark 2.3) the following lemma.Lemma 2.5. Let u be a continuous function on an open set A � Rn, andc � 0. Then the following are equivalent.(a) The function u is semiconcave with semiconcavity constant c.(b) For each x 2 A there exists � > 0 such that (3) holds whenever khk < �and [x� h; x+ h] � A.Proof. The implication (a)) (b) is trivial. Suppose that (b) holds. To prove(a), by Remark 2.3 it su�ces to prove that g(x) = u(x)� (c=2)kxk2 is locallyconcave on A. So, it is su�cient to prove that g is concave on each open convexC � A. Let x 2 C. Choose � > 0 by Lemma 2.5(b) and consider h 2 Rn withkhk < �. Then (2) and the choice of � give�2g(x+ h; x� h) = �2u(x+ h; x� h)� (c=2)khk2 � 0:Consequently, g is concave on C by Lemma 2.4(ii). �The notion of semiconcavity extends naturally to length spaces X. Theauthors working in the theory of length spaces use mostly the following termi-nology.De�nition 2.6. Let X be a geodesic space. Let G � X be open, c � 0, andf : G! R be a locally Lipschitz function.(i) We say that f is c-concave if, for each minimal curve : [a; b] ! G,the function g(t) = f � (t)� (c=2)t2 is concave on [a; b].(ii) We say that f is semiconcave on G if for each x 2 G there exists c � 0such that f is c-concave on an open neighbourhood of x.Remark 2.7. (i) If X = Rn, then c-concavity coincides with semicon-cavity with constant c.(ii) For the above terminology see [18, p. 5] or [19, p. 862].(iii) The function g in De�nition 2.6 is clearly concave if (f � )00(t) � c foreach t 2 (a; b).We will need the following simple characterization of c-concavity.

Properties of distance functions 5Lemma 2.8. Let Y be a geodesic space. Let M � Y be open, c � 0, andf :M ! R be a locally Lipschitz function. Then the following are equivalent.(i) f is c-concave on M .(ii) If x; y 2M , and s is an M-midpoint of x; y, then(4) f(x) + f(y)2 � f(s) � (c=2)d2;

where d := (1=2) dist(x; y).(iii) For each s 2M there exists � > 0 such that (4) holds, whenever x; y; dare as in (ii) and d < �.Proof. The implication (ii))(iii) is trivial. Now suppose that (i) holds. Toprove (ii), let x; y; s; d be as in (ii). Choose a minimal curve : [a; b] !M with (a) = x; (b) = y and ((1=2)(a + b)) = s. By (i), the functiong(t) = f � (t)� (c=2)t2 is concave on [a; b]. So ef := f � is semiconcave withsemiconcavity constant c on (a; b) by Remark 2.3. Consequently, �2 ef(b�h; a+h) � (c=2)j(1=2)(b� a)� hj2 for each 0 < h < (1=2)(b� a). By continuity ofef we clearly obtain (4), since d = (1=2)(b� a).To prove (iii))(i), consider a minimal curve : [a; b] ! M and supposethat f satis�es (iii). To prove (i), it is clearly su�cient (since f is continuous)to prove that(5) ef := f � is semiconcave on (a; b) with constant c:We will prove (5) applying Lemma 2.5 for A = (a; b) and u = ef . Let x 2 (a; b)be given. Choose � > 0 by applying (iii) to s := (x). Since � is clearlya witness for the validity of Lemma 2.5(b) (for A = (a; b) and u = ef), theassertion (5) follows. �

2.2. D.c. manifolds and d.c. surfaces.De�nition 2.9. Let C be a nonempty convex set in a real normed linear spaceX. A function f : C ! R is called d.c. (or DC, or \delta-convex") if it can berepresented as a di�erence of two continuous convex functions on C.If Y is a �nite-dimensional normed linear space, then a mapping F : C ! Yis called d.c., if y��F is a d.c. function on C for each linear functional y� 2 Y �.Remark 2.10. (i) To prove that F is d.c., it is clearly su�cient to showthat y� � F is d.c. for each y� from a basis of Y �.(ii) Each d.c. mapping is clearly locally Lipschitz.(iii) There are many works on optimization that deal with d.c. functions.A theory of d.c. (delta-convex) mappings in the case when Y is ageneral normed linear space was built in [23].

6 J. Rataj and L. Zaj���cekSome basic properties of d.c functions and mappings are contained in thefollowing lemma.

Lemma 2.11. Let X, Y , Z be �nite-dimensional normed linear spaces, letC � X be a nonempty convex set, and U � X and V � Y open sets.(a) ([1]) If the derivative of a function f on C is Lipchitz, then f is d.c.In particular, each a�ne mapping is d.c.(b) ([10]) If a mapping F : C ! Y is locally d.c. on C, then it is d.c. onC.(c) ([10]) Let a mapping F : U ! Y be locally d.c., F (U) � V , and letG : V ! Z be locally d.c. Then G � F is locally d.c. on U .(d) ([23]) Let F : U ! V be a bilipschitz bijection which is locally d.c. onU . Then F�1 is locally d.c on V .Since locally d.c. mappings are stable with respect to compositions (Lemma2.11(c)), the notion of an n-dimensional d.c. manifold can be de�ned in anobvious way. But the importance of this notion is clear only after Perelman's(unpublished) note [17]. As mentioned in [12], this notion was not probablystudied in the literature. However, d.c. manifolds in Euclidean spaces werestudied (independently on [17]) in the unpublished diploma thesis [6].

De�nition 2.12. Let X be a Hausdor� topological space and n 2 N.(i) We say that (U;') is an n-dimensional chart on X if U is a nonemptyopen subset of X and ' : U ! Rn a homeomorphism of U onto anopen set '(U) � Rn.(ii) We say that two n-dimensional charts (U1; '1) and (U2; '2) on X ared.c.-compatible if '1(U1)\'2(U2) = ; or '1(U1)\'2(U2) 6= ; and thetransition maps '2 � ('1)�1 and '1 � ('2)�1 are locally d.c. (on theirdomains '1(U1 \ U2) and '2(U1 \ U2), respectively).(iii) We say that a system A of n-dimensional charts on X is an n-dimen-sional d.c. atlas on X, if the domains of the charts from A cover Xand any two charts from A are d.c.-compatible.Obviously, each n-dimensional d.c. atlas A on X can be extended to auniquely determined maximal n-dimensional d.c. atlas (which consists of alln-dimensional charts on X that are d.c.-compatible with all charts from A).We will say that X is equipped with an (n-dimensional) d.c. structure (or witha structure of an n-dimensional d.c. manifold), if a maximal n-dimensional d.c.atlas on X is determined (e.g., by a choice of an n-dimensional d.c. atlas).Let f be a function de�ned on an open set G � X. Then we say that fis d.c. if f � '�1 is locally d.c. on '(U \ G) for each chart (U;') from themaximal d.c. atlas on X such that U \G 6= ;. Clearly, it is su�cient to checkthis condition for each chart from an arbitrary �xed d.c. atlas.

Properties of distance functions 7Remark 2.13. (i) If we consider, in the de�nition of the chart (U;'),a mapping ' from U to an n-dimensional unitary space H', thewhole De�nition 2.12 does not change sense. (This follows easily fromLemma 2.11 (a), (c).)(ii) If X; Y are nonempty spaces equipped with m;n-dimensional d.c.structures, respectively, then the Cartesian product X � Y is canoni-cally equipped with an (m+n)-dimensional d.c. structure. Indeed, letAX ;AY be m;n-dimensional d.c. atlases on X; Y , respectively. Then,

A = f(UX � UY ; 'X 'Y ) : (UX ; 'X) 2 AX ; (UY ; 'Y ) 2 AY gis an (m + n)-dimensional d.c. atlas on X � Y , if we de�ne ('X 'Y )(x; y) = ('X(x); 'Y (y)).(iii) If X; Y are equipped with m;n-dimensional d.c. structures, respec-tively, and f : X � Y ! R is d.c. then the section x 7! f(x; y) is d.c.on X for any y 2 Y , and the section y 7! f(x; y) is d.c. on Y for anyx 2 X.De�nition 2.14. Let H be an (n+ k)-dimensional unitary space (n; k 2 N).We say that a set M � H is a k-dimensional Lipschitz (resp. d.c.) surface, ifit is nonempty and for each x 2 M there exists a k-dimensional linear spaceQ � H, an open neighbourhoodW of x, a set G � Q open in Q and a Lipschitz(resp. locally d.c.) mapping h : G! Q? such that

M \W = fu+ h(u) : u 2 Gg:Remark 2.15. (i) Lipschitz surfaces were considered e.g. by Whitehead[26, p. 165] or Walter [25], who called them strong Lipschitz subman-ifolds. Obviously, each d.c. surface is a Lipschitz surface. For someproperties of d.c. surfaces see [29].(ii) If we suppose, in the above de�nition of a d.c. surface, that G is convexand h is d.c. and Lipschitz, we obtain clearly the same notion.(iii) Each Lipschitz (resp. d.c.) surface admits a natural structure of aLipschitz (resp. d.c.) manifold that is given by the charts of the form( �1;W \M), where (u) = u+ h(u); u 2 G.Lemma 2.16. Let Y be an n-dimensional unitary space (n � 2), t 2 Y ,fW � Y an open neighbourhood of t, and let g be a locally d.c. function on fWwhich is strictly di�erentiable at t with dg(t) 6= 0. Then there exists an openneighbourhood W � fW of t such that, for each r 2 R, the set

Lr := fx 2 W : g(x) = rgis either empty or an (n� 1)-dimensional d.c. surface in Y .

8 J. Rataj and L. Zaj���cekProof. Choose v 2 Y , kvk = 1, such that the directional derivative Dvg(t) isnonzero. Set E = v? and identify E � R with Y via the bijection (e; u) 7!e+ uv. Set '(e; u) := (e; g(e; u)); (e; u) 2 fW:It is easy to see that the mapping ' : fW ! Y is strictly di�erentiable at t and'0(t) is surjective. Thus by the well-known inverse mapping theorem (see e.g.[14, Theorem 1.60 and (1.15)]) there exists an open neighbourhood W � fW oft such that Z := '(W ) is open and ' is bilipschitz on W . By Lemma 2.11(d),the mapping '�1 is locally d.c. on Z. So '�1(e; u) = (e; h(e; u)); (e; u) 2 Z,for some locally d.c. function h on Z.Now suppose that r 2 R with Lr 6= ; is given. Consider an arbitraryy = (e1; r1) 2 Lr. Then '(y) = (e1; r) and we can choose �1 > 0, �2 > 0 suchthat U := B�1(e1)� (r � �2; r + �2) � Z. Then clearlyLr \ '�1(U) = f(e; u) : e 2 B�1(e1); u = h(e; r)g:Since '�1(U) is open and !(e) := h(e; r); e 2 B�1(e1); is a locally d.c. function,we have proved that Lr is an (n� 1)-dimensional d.c. surface in Y . �Lemma 2.17. Let H be an n-dimensional unitary space, V � H an openconvex set, and f : V ! Rm be a d.c. mapping. Then there exists a sequence(Ti) of (n�1)-dimensional d.c surfaces in H such that f is strictly di�erentiableat each point of V nS1i=1 Ti.Proof. Let f = (f1; : : : ; fm). By de�nition of a d.c. mapping, fj = �j � �j,where �j and �j are convex functions. By [27], for each j we can �nd asequence T jk , k 2 N, of (n� 1)-dimensional d.c. surfaces in H such that both�j and �j are di�erentiable at each point of Dj := H n S1k=1 T jk . Since eachconvex function is strictly di�erentiable at each point at which it is (Fr�echet)di�erentiable (see, e.g., [23, Proposition 3.8] for a proof of this well-knownfact), we conclude that each fj is strictly di�erentiable at each point of Dj.Since strict di�erentiablity of f clearly follows from strict di�erentiability ofall fj's, the proof is �nished after ordering all sets T jk , k 2 N, j = 1; : : : ;m, toa sequence (Ti). �

2.3. Convex surfaces.De�nition 2.18. A convex body in Rn is a compact convex subset with non-empty interior. Under a convex surface in Rn we understand the boundaryX = @C of a convex body C. A convex surface X is said to be polyhedral if itcan be covered by �nitely many hyperplanes.It is well known that a convex surface in Rn with its intrinsic metric is acomplete geodesic space with nonnegative curvature (see [4] or [3, x10.2]).

Properties of distance functions 9Obviously, each convex surface X is a d.c. surface (cf. Remark 2.20(iii)),and so has a canonical d.c. structure. In the following, we will work mainlywith \standard" d.c. charts of X.De�nition 2.19. Let X � Rn+1 be a convex surface. We say that (U;') isa standard n-dimensional chart on X, if there exist a unit vector e 2 Rn+1, aconvex, relatively open subset V of the hyperplane e?, and a Lipschitz convexfunction f : V ! R such that, setting F (x) := x + f(x)e; x 2 V , we haveU = F (V ) and ' = F�1. In this case we will say that (U;') is an (e; V )-standard chart on X and f will be called the convex function associated withthe standard chart.Remark 2.20. (i) Clearly, if (U;') is an (e; V )-standard chart on X and� denotes the orthogonal projection onto e?, then ' = � �U .(ii) Let (U1; '1) and (U2; '2) be standard charts as in the above de�nition.Then these charts are d.c.-compatible. Indeed, '�11 is a d.c. mappingfrom V1 to Rn+1 and '2 is a restriction of a linear mapping (see (i)).So '2 � ('1)�1 is locally d.c. by Lemma 2.11(a),(c).(iii) Let X � Rn+1 be a convex surface, z 2 X, and let C be the convexbody for which X = @C. Choose a 2 intC, set e := a�zka�zk and V :=�(B(a; �)), where � > 0 is su�ciently small and � is the orthogonalprojection of Rn+1 onto e?. Then it is easy to see that there exists an(e; V )-standard chart (U;') on X with z 2 U .By (ii) and (iii) above, the following de�nition is correct.De�nition 2.21. Let X � Rn+1 be a convex surface. Then the standard d.c.structure on X is determined by the atlas of all standard n-dimensional chartson X.Lemma 2.22. Let X � Rn+1 (n � 2) be a convex surface and let (U;') bean (e; V )-standard chart on X. Let T � e? be an (n � 1)-dimensional d.c.surface in e? with T \V 6= ;. Then '�1(T \V ) is an (n�1)-dimensional d.c.surface in Rn+1.Proof. Let f be the convex function associted with (U;'). Let z be an arbitrarypoint of '�1(T \ V ). Denote x := '(z). By De�nition 2.14 there exist an(n � 1)-dimensional linear space Q � e?, a set G � Q open in Q, an openneighbourhood W of x in e? and a locally d.c. mapping h : G ! Q?e? suchthat T \W = fu + h(u) : u 2 Gg. We can and will suppose that W � V .Observing that z 2 '�1(T \W ) and '�1(T \W ) is an open set in '�1(T \V ),'�1(T \W ) = fu+ h(u) + f(u+ h(u))e : u 2 Ggand u 7! h(u) + f(u + h(u))e is a locally d.c. mapping G ! Q?

Rn+1 , we �nishthe proof. �

10 J. Rataj and L. Zaj���cekLemma 2.23. (i) Let X be a convex surface in Rm. Then there exists asequence (Xk) of polyhedral convex surfaces in Rm converging to X inthe Hausdor� distance.(ii) Let convex surfaces Xk converge in the Hausdor� distance to a convexsurface X in Rm and let distX , distXk denote the intrinsic distanceson X, Xk, respectively. Assume that a; b 2 X, ak; bk 2 Xk, ak ! aand bk ! b. Then distXk(ak; bk)! distX(a; b).(iii) If Xk; X are as in (ii) then diamXk ! diamX, where diamXk; diamXis the intrinsic diameter of Xk; X, respectively.Proof. (i) is well known, see e.g. [21, x1.8.15].(ii) can be proved as in [3, Lemma 10.2.7], where a slightly di�erent assertionis shown. We present here the proof for completeness. Let C;Ck be convexbodies in Rm such that X = @C, Xk = @Ck, k 2 N, and assume, without lossof generality, that the origin lies in the interior of C. It is easy to show that,since the Hausdor� distance of X and Xk tends to zero, there exist k0 2 Nand a sequence "k & 0 such that(1� "k)C � Ck � (1 + "k)C; k � k0:For a convex body D in Rm and corresponding convex surface Y = @D, weshall denote by �Y the metric projection of Rm onto Y , de�ned outside of theinterior of D. The symbol distY denotes the intrinsic distance on the convexsurface Y . Let a; b; ak; bk from the assumption be given, and (for k � k0)denote eak = �Xk((1 + "k)a), ebk = �Xk((1 + "k)b). Since �Xk is a contraction(see e.g. [21, Theorem 1.2.2]), we have

distXk(eak;ebk) � dist(1+"k)X((1 + "k)a; (1 + "k)b)= (1 + "k)distX(a; b):Further, clearly eak ! a and ebk ! b, which implies that distXk(eak; ak)! 0 anddistXk(ebk; bk)! 0. Consequently,lim supk!1 distXk(ak; bk) � distX(a; b):The inequality lim infk!1 distXk(ak; bk) � distX(a; b) is obtained in a similarway, considering the metric projections of ak and bk onto (1� "k)X.(iii) is a straightforward consequence of (ii) and the compactness of X. �

Lemma 2.24. Let X � Rn+1 be a convex surface, (U;') an (e; V )-standardchart on X, and let f be the associated convex function. Let (Xk) be a sequenceof convex surfaces which tends in the Hausdor� metric to X, and W � V bean open convex set such that W � V . Then there exists k0 2 N such that, foreach k � k0, the surface Xk has an (e;W )-standard chart (Uk; 'k), and the

Properties of distance functions 11associated convex functions fk satisfy(6) fk(x)! f(x); x 2 W and lim supk!1 Lip fk � Lip f:Proof. Denote by C(Ck) the convex body for which X = @C (Xk = @Ck,respectively). Clearly, the convex function f has the formf(v) = infft 2 R : v + te 2 Cg; v 2 V:Let � be the orthogonal projection onto e? and denoteWr := fv 2 e? : dist(v;W ) < rg; r > 0:Let "; � > 0 be such that W"+� � V , and let k0 = k0(�) 2 N be such that theHausdor� distance of X and Xk (and, hence, also of C and Ck) is less that �for all k > k0. Fix a k > k0. It is easy to show thatf �k (v) = infft 2 R : v + te 2 Ckg; v 2 W"is a �nite convex function. We shall show that(7) jf �k (v)� f(v)j � (1 + Lip f)�; v 2 W":Take a point v 2 W" and denote x = v+ f(v)e 2 X and y = v+ f �k (v)e 2 Xk.From the de�nition of the Hausdor� distance, there must be a point c 2 Cwith kc� yk < �. This implies that for w := �(c) we have f(w) � c � e andf �k (v) = y � e � c � e� � � f(w)� � � f(v)� � Lip f � �:For the other inequality, note that, since f �k is convex, there exists a unit vectoru 2 Rn+1 with u � e =: �� < 0 such that (z� y) �u � 0 for all z 2 Ck (i.e., u isa unit outer normal vector to Ck at y). It is easy to see that (z� y) �u � � forall z 2 C, since the Hausdor� distance of C and Ck is less than �. Considerthe point z = w + f(w)e 2 C with w = v + �u�, where u� = �(u)=k�(u)k if�(u) 6= 0 and u� is any unit vector in e? if �(u) = 0. Then� � (z � y) � u = (w + f(w)e� v � f �k (v)e) � u= (w � v) � u+ (f(w)� f �k (v))(e � u)= �p1� �2 + (f(w)� f �k (v))(��)� �(1� �) + (f �k (v)� f(w))�;which implies that f �k (v) � f(w) + � � f(v) + � Lip f + �by the Lipschitz property of f , and (7) is veri�ed.We shall show now that for k > k0, Xk has an (e;W )-standard chart withassociated convex function fk := f �k � W (i.e., that fk is Lipschitz) and that(6) holds. Given two di�erent points u; v 2 W , we de�ne points u�; v� 2 W"as follows: we set u� = u � " v�ukv�uk , v� = v if fk(u) � fk(v), and u� = u,

12 J. Rataj and L. Zaj���cekv� = v + " v�ukv�uk if fk(u) � fk(v). Then, using (7) and convexity of f �k , weobtain jfk(u)� fk(v)jku� vk � jf �k (u�)� f �k (v�)jku� � v�k � Lip f + (2 + 2Lip f)�"whenever k > k0(�). Therefore, Lip fk � Lip f + (2+2Lip f)�" . Using this in-equality, (7), and the fact that � > 0 can be arbitrarily small, we obtain(6). �

3. Semiconcavity of distance functions on Alexandrov spacesLemma 3.1. Let H� be the hyperbolic plane of curvature � if � < 0 and theEuclidean plane if � = 0. Let � > 0 and a point p 2 H� be given. Then thefunction dist(�; p) is ��(�)-concave on G� := fx 2 H� : dist(x; p) > �g, where��(�) = � 1� ; � = 0;p�� coth(p���); � < 0:Proof. Let ' : [a; b]! G� be a minimal curve in H�. Then the image of ' is asubset of a line l; let f be the foot of the perpendicular from p to the line l (or pitself, if p lies on the line l). Let (t) (t 2 R) be an arc-length parametrizationof l such that (0) = f . Denote g(t) = dist( (t); p), a0 := �1('(a)) andb0 := �1('(b)). By Remark 2.7(iii) and monotonicity of ��, it is su�cient toshow that(8) g00(t) � ��(g(t)); t 2 (a0; b0):If p 2 l then g is clearly linear on (a0; b0) and the assertion is obvious. Soassume that g(0) > 0. If � = 0 then g(t)2 = t2 + g(0)2 and (8) follows easily.If � < 0, then we use the well-known formula (which follows, e.g., from [19,(3), p. 827])(9) cosh(p��g(t)) = cosh(p��g(0)) cosh(p��t); t 2 R:Di�erentiating, we obtainsinh(p��g(t)) � g0(t) = cosh(p��g(0)) � sinh(p��t)and

(10) p�� cosh(p��g(t))g0(t)2 + sinh(p��g(t))g00(t)= cosh(p��g(0)) cosh(p��t) = cosh(p��g(t));and deduceg00(t) = p��(cosh(p��g(t))� g0(t)2 cosh(p��g(t))sinh(p��g(t)) � p��(coth(p��g(t));as required. �

Properties of distance functions 13Theorem 3.2. Let X be a complete geodesic (Alexandrov) space with lowercurvature bound � � 0. Then the Cartesian product X2 with the product metric

dist((x1; x2); (y1; y2)) =qdist2(x1; y1) + dist2(x2; y2)is a complete geodesic space with lower curvature bound � as well, and thesquared distance g(x1; x2) := dist2(x1; x2) is c�-concave on M � X2, providedthat (i) M = X2 and c� = 18 if � = 0,(ii) M is open and bounded in X2, ! := sup(x1;x2)2M dist(x1; x2) and c� =10 + 8p��! coth(p��!) if � < 0.Proof. The assertion on the properties of X2 is well known, see e.g. [3, x3.6.1,x10.2.1].To prove (i) and (ii), we will show that g satis�es the condition (iii) ofLemma 2.8. To this end we assign to each s = (s1; s2) 2 M a corresponding� = �(s), distinguishing the cases s1 = s2 and s1 6= s2. If s1 = s2, we canchoose an arbitrary � > 0. In the case s1 6= s2 we set � := (1=4)dist(s1; s2)and(11) � = �(s) := min(��(�)�1; (1=3)�);where ��(�) is the number from Lemma 3.1. To verify Lemma 2.8 (iii), letx = (x1; x2) and y = (y1; y2) be arbitrary points in M and let s = (s1; s2) bean M -midpoint of x and y such thatd := (1=2)dist(x; y) < � = �(s):Note that si is a midpoint of xi and yi in X, i = 1; 2, see [19, x4.3]. ByLemma 2.8, it is su�cient to prove(12) g(x) + g(y)2 � g(s) + c�2 d2:Assume �rst that s1 = s2. Then, since dist(xi; yi) � dist(x; y) = 2d and s1is the midpoint of xi and yi (i = 1; 2), all the four points x1; y1; x2 and y2 havedistance from s1 less or equal to d . Hence,g(x) + g(y)2 = dist2(x1; x2) + dist2(y1; y2)2 � (2d)2 + (2d)22 = 4d2and (12) holds with c� = 8.If s1 6= s2, let t 2 X be a midpoint of s1 and s2 inX and consider comparisontriangles (cf. [3, pp. 107, 128]) ex1; ey1; et1 and ex2; ey2; et2 in the �-plane H� for thetriangles x1; y1; t and x2; y2; t, respectively. Let esi be the (unique) midpointof exi and eyi in H�, i = 1; 2. By Alexandrov-Toponogov theorem (called alsoToponogov's theorem, see [3, p. 360]) we obtain(13) dist(si; t) � dist(esi; eti); i = 1; 2:

14 J. Rataj and L. Zaj���cekNow we will estimate dist(esi; eti) from below using Lemma 3.1 (used forp := eti) and Lemma 2.8(ii). Lemma 3.1 gives that dist(�; eti) is ��(�)-concaveon Ai� := fx 2 H� : dist(x; eti) > �g. By de�nition of esi, there exists a minimalcurve 'i in H� with endpoints exi and eyi such that esi is a midpoint of 'i. Foreach u from the image of 'i, we have

(14) dist(u; eti) � dist(exi; eti)� dist(exi; u) � dist(xi; t)� dist(xi; yi)� (dist(si; t)� d)� 2d = 2�� 3d > �:So, esi is an Ai�-midpoint of exi and eyi, and therefore Lemma 2.8(ii) gives(15) dist(esi; eti) � dist(exi; eti) + dist(eyi; eti)2 � ��(�)2

�dist(exi; eyi)2�2 ;

i = 1; 2. Using the de�nitions of exi, eyi and esi, (13) and (15), we obtaindist(s1; s2) = dist(s1; t) + dist(t; s2)� dist(x1; t) + dist(y1; t) + dist(x2; t) + dist(y2; t)2 � ��(�)2 d2� dist(x1; x2) + dist(y1; y2)2 � ��(�)2 d2:

Note that�dist(x1; x2) + dist(y1; y2)2�2

= dist2(x1; x2) + dist2(y1; y2)2 � (dist(x1; x2)� dist(y1; y2))24� g(x) + g(y)2 � 4d2;since jdist(x1; x2)� dist(y1; y2)j � dist(x1; y1) + dist(x2; y2) � 4d;hence,g(x) + g(y)2 � 4d2 � �dist(s1; s2) + ��(�)2 d2�2

� g(s) + 2dist(s1; s2)��(�)2 d2 + ���(�)2�2 d4

� g(s) + 4���(�)d2 + d2:We have used the de�nition of � and the assumption d < ��(�)�1 in the lastestimate. Since the function r��(r) is non-decreasing on (0;1) and � � !, we

Properties of distance functions 15obtain g(x) + g(y)2 � g(s) + (5 + 4!��(!))d2:It remains to substitute the form of ��(!) from Lemma 3.1 to obtain (12). �

Remark 3.3. Results similar to Theorem 3.2 can be found in the literature.Mantegazza and Mennucci [15, Proposition 3.4] show the local semiconcav-ity of the distance function from a closed subset of a Riemannian manifold.Plaut [19, Proposition 125] considers the distance from a closed subset in anAlexandrov space. Villani [24, Example 10.22] shows that the squared dis-tance function in a Riemannian manifold of nonnegative sectional curvature isuniformly semiconcave in both variables.4. Extrinsic properties of distance functions on convexsurfacesLemma 4.1. Let X be a polyhedral convex surface in Rn+1, T 2 X, and (U;')be an (e; V )-standard chart on X such that T 2 U . Let f be the associatedconvex function and t := '(T ). Then there exists a � > 0 such that for allx; y 2 V with t = (x+ y)=2 and kx� tk = ky � tk < � we have

dist(S; T ) � 2�2f(x; y);whenever S is a midpoint of '�1(x); '�1(y).Proof. Denoting F := '�1, we have F (u) = u+ f(u)e. Let L be the Lipschitzconstant of f . It is easy to see that we can choose �0 > 0 such that forany x 2 V with kx � tk < �0, the function f is a�ne on the segment [x; t].Then we take � � �0=L, such that for any two points x; y 2 B(t; �), anyminimal curve connecting F (x) and F (y) (and, hence, also any midpoint ofF (x); F (y)) lies in U . Let two points x; y 2 B(t; �) with t = x+y2 be given,denote � = �2f(x; y) and let S be a midpoint of F (x); F (y) (lying necessarilyin U), and set s = '(S). Note that � � L�.From the parallelogram law, we obtain

2kF (x)� Tk2 + 2kF (y)� Tk2 = kF (y)� F (x)k2 + 4�2;since(16) � = F (x) + F (y)2 � T :Taking the square root, and using the inequality a+b � p2a2 + 2b2, we obtain

kF (x)� Tk+ kF (y)� Tk �pkF (y)� F (x)k2 + 4�2:

16 J. Rataj and L. Zaj���cekIt is clear that the geodesic distance of F (x) and F (y) is at most kF (x)�Tk+kF (y)� Tk (which is the length of a curve in X connecting F (x) and F (y)).Thus,kS�F (x)k � dist(S; F (x)) = 12dist(F (x); F (y)) �

s�kF (y)� F (x)k2�2 +�2

and the same upper bound applies to kS � F (y)k. Summing the squares ofboth distances, we obtainkS � F (x)k2 + kS � F (y)k2 � 12kF (y)� F (x)k2 + 2�2

and, since the left hand side equals, again by the parallelogram law,12 �kF (y)� F (x)k2 + k2S � (F (x) + F (y)k2� ;we arrive at(17) S � F (x) + F (y)2

� �:Considering the orthogonal projections of S and F (x)+F (y)2 onto e?, we obtainks� tk � � � L� � �0and, hence, we have dist(S; T ) = kS � Tk;since f is a�ne on [s; t]. On the other hand, equations (16) and (17) implykS � Tk � 2�, which completes the proof. �

Proposition 4.2. Let X � Rn+1 be a convex surface and let (Ui; 'i) be (ei; Vi)standard charts, i = 1; 2. Let f1, f2 be the corresponding convex functions. Setg(x1; x2) = dist2('�11 (x1); '�12 (x2)); x1 2 V1; x2 2 V2;where dist is the intrinsic distance on X. Then the function g�c�d is concaveon V1 � V2, where c(x1; x2) = 18(1 + L2)(kx1k2 + kx2k2);d(x1; x2) = 4M(f1(x1) + f2(x2));L = maxfLip f1;Lip f2g and M is the intrinsic diameter of X.Proof. Assume �rst that the convex surface X is polyhedral. We shall showthat for any t 2 V1 � V2 there exists � > 0 such that(18) �2g(x; y) � �2c(x; y) + �2d(x; y)

Properties of distance functions 17for all x; y 2 B(t; �) � V1�V2 with t = (x+ y)=2, which implies the assertion,see Lemma 2.4. We have

�2g(x; y) = g(x) + g(y)2 � g(t)= �g(x) + g(y)2 � g(s)�+ (g(s)� g(t)) ;

whenever s = (s1; s2) 2 V1 � V2 is such that ('�11 (s1); '�12 (s2)) is a midpointof ('�11 (x1); '�12 (x2)) and ('�11 (y1); '�12 (y2)) in X2, where x = (x1; x2) andy = (y1; y2). By Theorem 3.2 and Lemma 2.8(ii), the �rst summand is boundedfrom above by9 dist2('�11 (x1); '�11 (y1)) + dist2('�12 (x2); '�12 (y2))4 :

Since clearlydist('�1i (xi); '�1i (yi)) �p1 + (Lip fi)2kxi � yik; i = 1; 2;we getg(x) + g(y)2 � g(s) � 92(2 + (Lip f1)2 + (Lip f2)2)kx1 � y1k2 + kx2 � y2k22� �2c(x; y)(we use the fact that �2c(x; y) = 18(1+L2)(kx� yk=2)2, see (2)). In order toverify (18), it remains thus to show that(19) jg(s)� g(t)j � �2d(x; y):Denote t = (t1; t2), s = (s1; s2), Ti = '�1i (ti) and Si = '�1i (si), i = 1; 2 . Wehave jg(s)� g(t)j = jdist2(S1; S2)� dist2(T1; T2)j� 2M jdist(S1; S2)� dist(T1; T2)j� 2M(dist(S1; T1) + dist(S2; T2));where the last inequality follows from the (iterated) triangle inequality. Ap-plying Lemma 4.1, we get dist(Si; Ti) � 2�2fi(xi; yi), i = 1; 2, for � su�cientlysmall (remind that Si is the midpoint of '�1i (xi); '�1i (yi), see [19, x4.3]). Sinceclearly �2d(x; y) = 4M(�2f1(x1; y1) + �2f2(x2; y2));(19) follows.Let nowX be an arbitrary convex surface. Let (Xk) be a sequence of polyhe-dral convex surfaces which tends in the Hausdor� metric to X. Consider arbi-trary open convex sets Wi � Vi with Wi � Vi, i = 1; 2. Applying Lemma 2.24(and considering a subsequence of Xk if necessary), we �nd (ei;Wi)-standard

18 J. Rataj and L. Zaj���cekcharts (Ui;k; 'i;k) of Xk such that the associated convex functions fi;k convergeto fi �Wi , L�i := limk!1 Lip fi;k exists and L�i � Lip fi, i = 1; 2.By the �rst part of the proof we know that the function k(x1; x2) := gk(x1; x2)� 18(1+L2k)(kx1k2+kx2k2)� 4Mk(f1;k(x1)+ f2;k(x2));where Mk is the intrinsic diameter of Xk and Lk = max(Lip f1;k;Lip f1;k),is concave on W1 � W2. Obviously, Lk ! L� := max(L�1; L�2) � L andLemma 2.23 implies that gk ! g and Mk !M . Consequently,limk!1 k(x1; x2) = g(x1; x2)� 18(1+L�2)(kx1k2+ kx2k2)� 4M(f1(x1)+ f2(x2))is concave on W1�W2. Since L� � L , we obtain that g� c� d is concave onW1 �W2. Thus g � c� d is locally concave, and so concave, on V1 � V2. �Proposition 4.2 has the following immediate corollary (recall the de�nitionof a d.c. function on a d.c. manifold, De�nition 2.12, and the de�nition of thed.c. structure on X2, Remark 2.13 (ii)).Theorem 4.3. Let X be a convex surface in Rn+1. Then the squared distancefunction (x; y) 7! dist2(x; y) is d.c. on X2.Using Remark 2.13 (iii), we obtainCorollary 4.4. Let X be a convex surface in Rn+1 and let x0 2 X be �xed.Then the squared distance from x0, x 7! dist2(x; x0), is d.c. on X.Since the function g(z) = pz is d.c. on (0;1), Lemma 2.11(c) easily impliesCorollary 4.5. Let X be a convex surface in Rn+1 and let x0 2 X be �xed.Then the distance from x0, x 7! dist(x; x0), is d.c. on X n fx0g.Remark 4.6. If n = 1, it is not di�cult to show that the function x 7!dist(x; x0) is d.c. on the whole X. On the other hand, we conjecture that thisstatement is not true in general for n � 2.

5. ApplicationsOur results on distance functions can be applied to a number of problemsfrom the geometry of convex surfaces that are formulated in the language ofdistance functions. We present below applications concerning r-boundaries(distance spheres), the multijoined locus, and the ambiguous locus (exoskele-ton) of a closed subset of a convex surface. Recall that r-boundaries and am-biguous loci were studied (in Euclidean, Riemannian and Alexandrov spaces)in a number of articles (see, e.g., [9], [22], [30], [11]).Theorem 5.1. Let X � Rn+1 be a convex surface and ; 6= F � X a closedset. Denoting dF := dist(�; F ),(i) the function (dF )2 is d.c. on X and

Properties of distance functions 19(ii) the function dF is d.c. on X n F .Proof. Since X is compact, we can choose a �nite system (Ui; 'i), i 2 I, of(ei; Vi)-standard charts which forms a d.c. atlas on X. Let fi, i 2 I, be thecorrresponding convex functions. Choose L > 0 such that Lip fi � L for alli 2 I and let M be the intrinsic diameter of X. To prove (i), it is su�cient toshow that, for all i 2 I, (dF )2 � ('i)�1 is d.c. on Vi. So �x i 2 I and consideran arbitrary y 2 F . Choose j 2 I with y 2 Uj. Set!(x) := 18(1 + L2)kxk2 + 4Mfi(x); x 2 Vi:Proposition 4.2 (used for '1 = 'i and '2 = 'j) easily implies that the functionhy(x) = dist2('�1i (x); y)� !(x) is concave on Vi. Consequently, the function (x) := (dF )2 � ('i)�1(x)� !(x) = infy2F hy(x)is concave on Vi. So (dF )2 � ('i)�1 = + ! = ! � (� ) is d.c. on Vi. Thus(i) is proved. Since the function g(z) = pz is d.c. on (0;1), Lemma 2.11(c)easily implies (ii). �Theorem 5.2. Let X � Rn+1 (n � 2) be a convex surface and ; 6= K � X aclosed set. For r > 0, consider the r-boundary Kr := fx 2 X : dist(x;K) =rg. (i) There exists a set G � X nK which is open and dense in X nK anda sequence (Si); i 2 N; of (n � 1)-dimensional d.c. surfaces in Rn+1such that Si � X, X nK � G [S1i=1 Si and, for each r > 0, Kr \Gis either empty, or an (n� 1)-dimensional d.c. surface.(ii) For each r > 0, the set Kr can be covered by countably many (n� 1)-dimensional d.c. surfaces lying in X.(iii) There exists a �rst category set N � (0;1) such that, for each r 2(0;1) nN , either Kr = ; or there exists an (n� 1)-dimensional d.c.surface which is a dense subset of Kr.Proof. Choose a system (Ui; 'i), i 2 N, of (ei; Vi)-standard charts on X suchthat X nK = S1i=1 Ui. Let fi, i 2 N, be the corrresponding convex functions.By Theorem 5.1, we know that di := dK � '�1i is locally d.c. on Vi, wheredK := dist(�; K). Then di is d.c. on Vi by Lemma 2.11(b). So, for each i, we can�nd by Lemma 2.17 a sequence T ik, k 2 N, of (n� 1)-dimensional d.c. surfacesin e?i such that di is strictly di�erentiable at each point of Zi := Vi nS1k=1 T ik.Order all nonempty subsets of Rn+1 that are of the form '�1i (Vi \ T ik); i 2I; k 2 N, to a sequence (Sj)1j=1. By Lemma 2.22, each Sj is an (n � 1)-dimensional d.c. surface in Rn+1. Fix a point x 2 H := X n (K [ S1j=1 Sj)and choose i 2 N with x 2 Ui. We know that di is strictly di�erentiableat t = 'i(x). Moreover, t is not a stationary (critical) point of di (i.e., thedi�erential of di at t is nonzero). Indeed, otherwise there exists � > 0 such

20 J. Rataj and L. Zaj���cekthat jdi(�)� di(t)j < k� � tk whenever k� � tk < �. Choose a minimal curve with endpoints x and u 2 K and length s = dist(x;K). Choosing a pointx� on the image of which is su�ciently close to x and putting � := 'i(x�),we clearly have k� � tk < � and jdi(�)� di(t)j = dist(x; x�) � k� � tk, whichis a contradiction.Now Lemma 2.16 and Lemma 2.22 easily imply that there exists �x > 0such that B(x; �x) \K = ; and B(x; �x) \Kr is either empty, or an (n� 1)-dimensional d.c. surface for each r > 0. So, setting G := Sx2H B(x; �x) weconclude the proof of (i).The condition (ii) follows immediately from (i).Also (iii) follows from (i) by standard topological arguments. Since Kr \Gis either empty, or an (n� 1)-dimensional d.c. surface, it is su�cient to provethat N := fr 2 (0;1) : Kr \G is not dense in Krgis a �rst category set. Choose a basis (Bm); m 2 N; of open subsets of X nKand denote Nm;i := fr 2 (0;1) : ; 6= Kr \Bm � Sig:It is su�cient to prove that(20) N � 1[

m;i=1 Nm;iand(21) each set Nm;i is nowhere dense:To prove (20), �x an arbitrary r 2 N . Then the closed setKrnG is not nowheredense in the complete space Kr, and so Kr n G is of the second category inKr. Since Kr nG � S1i=1 Si, there exists i� such that Si� is not nowhere densein Kr, and consequently Si� is dense in a nonempty relatively open subset ofKr. Since Si� is clearly locally compact, we can easily �nd m� 2 N such that; 6= Kr \Bm� � Si� , i.e. r 2 Nm�;i� .To prove (21), suppose to the contrary that some Nm;i is dense in an interval(r1; r2). Choose r 2 (r1; r2) \ Nm;i and x 2 Bm \ Kr � Si. Further choose! > 0 so small that (r�!; r+!) � (r1; r2) and B(x; !) � Bm. We shall showthat Si is dense in B(x; !). Consider an arbitrary y 2 B(x; !) and choosea minimal curve with endpoints y and f 2 K and length ` = dist(y;K).Clearly ` 2 (r1; r2). So, for each " > 0, we can �nd r0 2 (r1; r2) \ Nm;i and(on the image of ) a point z 2 B(y; ") \Kr0 \ Bm � Si. Hence Si is densein B(x; !). So, since Si is locally compact, Si contains a ball B�, which is acontradiction, since the Hausdor� dimensions of Si and B� are n � 1 and n,respectively. �

Properties of distance functions 21Remark 5.3. The r-boundaries of sets in Euclidean spaces were studied in anumber of articles. In this context, the property (ii) is contained in [7].If K is a closed subset of a length space X, the multijoined locus M(K) ofK is the set of all points x 2 X such that the distance from x to K is realizedby at least two di�erent minimal curves in X. If two such minimal curvesexist that connect x with two di�erent points of K, x is said to belong to theambiguous locus A(K) of K. The ambiguous locus of K is also called skeletonof X nK (or exoskeleton of K, [11]).Zam�rescu [30] studies the multijoined locus in a complete geodesic (Alexan-drov) space of curvature bounded from below and shows that it is �-porous.An application of Theorem 5.1 yields a stronger result for convex surfaces:Theorem 5.4. Let K be a closed subset of a convex surface X � Rn+1 (n � 2).ThenM(K) (and, hence, also A(K)) can be covered by countably many (n�1)-dimensional d.c. surfaces lying in X.Proof. Let (U;') be an (e; V )-standard chart on X. It is clearly su�cient toprove that M(K)\ U can be covered by countably many (n� 1)-dimensionald.c. surfaces. Set F := '�1 and denote by dK(z) the intrinsic distance ofz 2 X from K. Since both the mapping F and the function dK � F are d.c.on V (see Theorem 5.1 and Lemma 2.11), they are by Lemma 2.17 strictlydi�erentiable at all points of V nN , where N is a countable union of (n� 1)-dimensional d.c. surfaces in e?. By Lemma 2.22, F (N \ V ) is a countableunion of (n� 1)-dimensional d.c. surfaces in Rn+1. So it is su�cient to provethatM(K)\U � F (N). To prove this inclusion, suppose to the contrary thatthere exists a point x 2 M(K) \ U such that both F and dK � F are strictlydi�erentiable at x.We can assume without loss of generality that x = 0. Let T := (dF (0))(e?)be the vector tangent space to X at 0. Let P be the projection of Rn+1 ontoT in the direction of e and de�ne Q := (P �U)�1. It is easy to see thatQ = F � (dF (0))�1 and therefore dQ(0) = (dF (0)) � (dF (0))�1 = idT .Since 0 2 M(K), there exist two di�erent minimal curves �; : [0; r] ! Xsuch that r = dK(0), �(0) = (0) = 0, �(r) 2 K, and (r) 2 K. As anyminimal curves on a convex surface, � and have right semitangents at 0(see [4, Corollary 2]); let u; v 2 Rn+1 be unit vectors from these semitangents.Further, [13, Theorem 2] easily implies that u 6= v.Clearly dK � �(t) = r � t; t 2 [0; r], and (P � �)0+(0) = P (�0+(0)) = u.Further observe that dK �Q is di�erentiable at 0, since dK �F is di�erentiableat 0 = (dF (0))�1(0). Using the above facts, we obtain(d(dK �Q)(0))(u) = (d(dK �Q)(0))((P � �)0+(0)) = (dK �Q � P � �)0+(0)= (dK � �)0+(0) = �1:In the same way we obtain (d(dK �Q)(0))(v) = �1.

22 J. Rataj and L. Zaj���cekThus, u+ v 6= 0 and, by the linearity of the di�erential,

(d(dK �Q)(0))� u+ vku+ vk� = �2ku+ vk < �1:

Thus there exists " > 0 such that(22) kd(dK �Q)(0)k > 1 + ":Since dQ(0) = idT and Q = F � (dF (0))�1 is clearly strictly di�erentiable at0, there exists � > 0 such thatkQ(p)�Q(q)� (p� q)k � "kp� qk; p; q 2 B(0; �) \ T;and consequently Q is Lipchitz on B(0; �) \ T with constant 1 + ". Let p; q 2B(0; �) \ T and consider the curve ! : [0; 1] ! X, !(t) = Q(tp + (1 � t)q).Then clearly dist(Q(p); Q(q)) � length ! � (1 + ")kp� qk:ConsequentlykdK �Q(p)� dK �Q(q)k � dist(Q(p); Q(q)) � (1 + ")kp� qk:Thus the function dK �Q is Lipchitz on B(0; �)\T with constant 1+ ", whichcontradicts (22). �Remark 5.5. An analoguous result on the ambiguous loci in a Hilbert spacewas proved in [28].6. Remarks and questionsThe results of [17] and Corollary 4.5 suggest that the following de�nition isnatural.De�nition 6.1. Let X be a length space and let an open set G � X beequipped with an n-dimensional d.c. structure. We will say that this d.c.structure is compatible with the intrinsic metric on X, if the following hold.(i) For each d.c. chart (U;'), the map ' : (U; dist) ! Rn is locallybilipschitz.(ii) For each x0 2 X, the distance function dist(x0; �) is d.c. (with respectto the d.c. structure) on G n fx0g.If M is an n-dimensional Alexandrov space with curvature bounded frombelow, the results of [17] give that there exists an open dense set M� � Mwith dimH(M n M�) � n � 2 and an n-dimensional d.c. structure on M�compatible with the intrinsic metric on M (cf. [17, p. 6, line 9 from below]).Since the components of each chart of this d.c. structure are formed by distancefunctions, Lemma 2.11(d) easily implies that no other d.c. structure on M�which is compatible with the intrinsic metric exists.

Properties of distance functions 23Let X � Rn+1 be a convex surface. Then Corollary 4.5 gives that thestandard d.c. structure on X is compatible with the intrinsic metric on X.By the above observations, there is no other compatible d.c. structure on the(open dense) \Perelman's set" X�. We conjecture that this uniqueness is truealso on the whole X. Further note that the standard d.c. structure on X hasan atlas such that all coresponding transition maps are C1. Indeed, let C bethe convex body for which X = @C. We can suppose 0 2 intC and �nd r > 0such that B(0; r) � intC. Now \identify" X with the C1 manifold @B(0; r)via the radial projection of X on @B(0; r). Then, this bijection transfers theC1 structure of @B(0; r) on X.We conclude with the following problem.Problem Let f : Rn ! R be a semiconcave (resp d.c.) function. Con-sider the \semiconcave surface" (resp. d.c. surface) X := graph f equippedwith the intrinsic metric. Let x0 2 X. Is it true that the distance functiondist(x0; �) is d.c. on X n fx0g with respect to the natural d.c. structure (givenby the projection onto Rn)? In other words, is the natural d.c. structure onX compatible with the intrinsic metric on X?

Remark 6.2. (i) If f is convex, then Corollary 4.5 easily implies (con-sidering a convex surface eX which contains su�ciently large part ofX) that the answer is positive. Moreover, denoting x� := (x; f(x)) forx 2 Rn, Theorem 4.3 gives that the function g(x; y) := dist2(x�; y�) isd.c. on R2n.(ii) If f is semiconcave, then each minimal curve ' on X has bounded turnin Rn+1 by [20]. Thus some interesting results on intrinsic propertiesextend from convex surfaces to the case of semiconcave surfaces. So,there is a chance that the above problem has the a�rmative answerin this case. However, we were not able to extend our proof to thiscase.References[1] A. D. Alexandrov, On surfaces represented as the di�erence of convex functions, Izv.Akad. Nauk. Kaz. SSR 60, Ser. Math. Mekh. 3 (1949), 3{20 (in Russian).[2] , Surfaces represented by the di�erences of convex functions, Dokl. Akad. NaukSSSR (N.S.) 72 (1950), 613{616 (in Russian).[3] D. Burago, Y. Burago, S. Ivanov, A course in metric geometry, Graduate Studies inMathematics, Volume 33, Amer. Math. Soc., Providence, 2001.[4] S.V. Buyalo, Shortest paths on convex hypersurfaces of Riemannian spaces, Zap. Nau�cn.Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 66 (1976), 114{132.[5] P. Cannarsa, C. Sinestrari, Semiconcave functions, Hamilton-Jacobi equations, andoptimal control, Progress in Nonlinear Di�erential Equations and their Applications,58. Birkh�auser Boston, Inc., Boston, MA, 2004.

24 J. Rataj and L. Zaj���cek[6] J. Duda, Delta convex mappings, diploma thesis (in Czech), Charles University, Prague,2000.[7] J. Duda, Implicit functions and properties of distance function, unpublished preprint,2008.[8] G. Durand, Sur une g�en�eralisation des surfaces convexes, Journ. de Math. (9) 10 (1931),335-414.[9] S. Ferry, When "-boundaries are manifolds, Fund. Math. 90 (1976), 199{210.[10] P. Hartman, On functions representable as a di�erence of convex functions, Paci�c J.Math. 9 (1959), 707{713.[11] D. Hug, G. Last, W. Weil, A local Steiner-type formula for general closed sets andapplications, Math. Z. 246 (2004), 237{272.[12] V. Kapovitch, Perelman's stability theorem, preprint (2007), arXiv:math/o703002v3.[13] A.D. Milka, Shortest lines on convex surfaces (Russian), Dokl. Akad. Nauk SSSR 248(1979), 34{36.[14] B.S. Mordukhovich, Variational analysis and generalized di�erentiation I., Basic the-ory, Grundlehren der Mathematischen Wissenschaften 330, Springer-Verlag, Berlin,2006.[15] C. Mantegazza, A.C. Mennucci, Hamilton-Jacobi equations and distance functions onRiemannian manifolds, Appl. Math. Optim. 47 (2003), 1{25.[16] Y. Otsu, T. Shioya, The Riemann structure of Alexandrov spaces, J. Di�erential Geom.39 (1994), 629{658.[17] G. Perelman, DC structure on Alexandrov space, an unpublished preprint (1995), avail-able at http://www.math.psu.edu/petrunin/papers/papers.html.[18] A. Petrunin, Semiconcave functions in Alexandrov's geometry, an unpublished preprint(2007), available at http://ftp.math.psu.edu/petrunin/papers/grove/convexity.pdf.[19] C. Plaut, Metric spaces of curvature � k, Handbook of geometric topology, 819{898,North-Holland, Amsterdam, 2002.[20] Yu. G. Reshetnyak, On a generalization of convex surfaces (Russian), Mat. Sbornik 40(82)(1956), 381{398.[21] R. Schneider, Convex bodies: the Brunn-Minkowski theory, Cambrigde University Press,Cambrigde, 1993.[22] K. Shiohama, M. Tanaka, Cut loci and distance spheres on Alexandrov surfaces, Actesde la Table Ronde de G�eom�etrie Di��erentielle (Luminy, 1992), 531{559, S�emin. Congr.,1, Soc. Math. France, Paris, 1996.[23] L. Vesel�y, L. Zaj���cek, Delta-convex mappings between Banach spaces and applications,Dissertationes Math. (Rozprawy Mat.) 289 (1989), 52 pp.[24] C. Villani, Optimal transport, old and new, Springer, Berlin, 2009.[25] R. Walter, Some analytical properties of geodesically convex sets, Abh. Math. Sem.Univ. Hamburg 45 (1976), 263{282.[26] J.H.C. Whitehead, Manifolds with transverse �elds in Euclidean space, Ann. Math. 73(1961), 154{212.[27] L. Zaj���cek, On the di�erentiation of convex functions in �nite and in�nite dimensionalspaces, Czechoslovak Math. J. 29 (1979), 292{308.[28] L. Zaj���cek, Di�erentiability of the distance function and points of multi-valuedness ofthe metric projection in Banach space, Czechoslovak Math. J. 33 (1983), 340{348.[29] L. Zaj���cek, On Lipschitz and d.c. surfaces of �nite codimension in a Banach space,Czechoslovak Math. J. 58 (2008), 849{864.

Properties of distance functions 25[30] T. Zam�rescu, On the cut locus in Alexandrov spaces and applications to convex sur-faces, Paci�c J. Math. 217 (2004), 375{386.Charles University, Faculty of Mathematics and Physics, Sokolovsk�a 83,186 75 Praha 8, Czech RepublicE-mail address: rataj@karlin.mff.cuni.czE-mail address: zajicek@karlin.mff.cuni.cz