Ball and Spindle Convexity with respect to a Convex Body
BBall and Spindle Convexity with respect toa Convex Body
Zsolt L´angi, M´arton Nasz´odi and Istv´an Talata
Abstract.
Let C ⊂ R n be a convex body. We introduce two notions ofconvexity associated to C. A set K is C - ball convex if it is the inter-section of translates of C , or it is either ∅ , or R n . The C -ball convexhull of two points is called a C -spindle. K is C - spindle convex if it con-tains the C -spindle of any pair of its points. We investigate how somefundamental properties of conventional convex sets can be adapted to C -spindle convex and C -ball convex sets. We study separation proper-ties and Carath´eodory numbers of these two convexity structures. Weinvestigate the basic properties of arc-distance, a quantity defined bya centrally symmetric planar disc C , which is the length of an arc of atranslate of C , measured in the C -norm, that connects two points. Thenwe characterize those n -dimensional convex bodies C for which every C -ball convex set is the C -ball convex hull of finitely many points. Finally,we obtain a stability result concerning covering numbers of some C -ballconvex sets, and diametrically maximal sets in n -dimensional Minkowskispaces. Mathematics Subject Classification (2010).
Keywords. ball convexity, spindle convexity, ball-polyhedron, separation,Carath´eodory’s theorem, convexity structure, illumination, arc-distance.
1. Introduction
Closed convex sets may be introduced in two distinct manners: either asintersections of half-spaces, or as closed sets which contain the line segmentsconnecting any pair of their points. We develop these approaches further to
The first named author was supported by the J´anos Bolyai Research Scholarship of theHungarian Academy of Sciences.The second named author was partially supported by the Hung. Nat. Sci. Found. (OTKA)grants: K72537 and PD104744.The third named author was partially supported by the Hung. Nat. Sci. Found. (OTKA)grant no. K68398. a r X i v : . [ m a t h . M G ] S e p Zsolt L´angi, M´arton Nasz´odi and Istv´an Talataobtain the notions of ball convexity and spindle convexity with respect to aconvex body. Let C be a convex body (a compact convex set with non-emptyinterior) in Euclidean n -space, R n . Definition 1.1.
A set K ⊂ R n is called ball convex with respect to C (shortly, C -ball convex ), if it is either ∅ , or R n , or the intersection of all translates of C that contain K . Definition 1.2.
Consider two (not necessarily distinct) points p, q ∈ R n suchthat there is a translate of C that contains both p and q . Then the C -spindle (denoted as [ p, q ] C ) of p and q is the intersection of all translates of C thatcontain p and q . If no translate of C contains p and q , we set [ p, q ] C = R n .We call a set K ⊂ R n spindle convex with respect to C (shortly, C -spindleconvex ), if for any p, q ∈ K , we have [ p, q ] C ⊂ K . In this paper, we study the geometric properties of ball and spindleconvex sets depending on C , and how these two families are related to eachother.In 1935, Mayer [29] defined the notion of “ ¨Uberkonvexit¨at”. His defini-tion coincides with our spindle convexity with the additional assumption that C is smooth and strictly convex. Attributing the concept to [30], Valentine(p.99 of [40]) uses the term “ r -convex” for ball convex sets in a Minkowskispace (where the radius of the intersecting balls is r ). For other generalizednotions of convexity related to Minkowski spaces, see [27].Intersections of (finitely many) Euclidean unit balls on the plane werestudied by Bieberbach [7], and in three dimensions by Heppes [14], Heppesand R´ev´esz [15], Straszewitcz [39] and Gr¨unbaum [13]. Recent developmentshave led to further investigations of sets that are ball convex with respect tothe Euclidean unit ball B n . For these results, the reader is referred to [4],[6], [8], [18], [19], [20] and [32]. A systematic investigation of these sets wasstarted by Bezdek et al. [5]: the authors examined how fundamental proper-ties of convex sets can be transferred to sets that are ball convex with respectto B n ; in particular, they gave analogues of Kirchberger’s and Carath´eodory’stheorems, examined separation properties of ball convex sets and variants ofthe Kneser-Poulsen conjecture. It is also shown there that the notions of balland spindle convexity coincide when C = B n . In the spirit of the results in[5], in [21] generalizations of Kirchberger’s theorem are proved regarding sep-aration of finite point sets by homothets or similar copies of a given convexregion. Using ball-polyhedra, the authors of [18] give a characterization offinite sets in Euclidean 3-space for which the diameter of the set is attaineda maximal number of times. The notion of C -ball convex hull (cf. Defini-tion 2.2) was defined for centrally symmetric plane convex bodies by Martiniand Spirova [26].In Section 2, we introduce our notation and basic concepts. In Section 3,we examine how the notions of ball and spindle convexity are related to eachother and how sets are separated by translates of C . In Section 4, we definethe arc-distance between two points with respect to a planar disc C , andall and Spindle Convexity with respect to a Convex Body 3examine when the triangle inequality holds and when it fails. In Section 5,we introduce and examine the Carath´eodory numbers associated to theseconvexity notions. In Section 6, we give a partial characterization of convexbodies C for which every C -ball convex set is the C -ball convex hull of finitelymany points. Finally, in Section 7, we prove that the operation of takingintersections of translates of C is stable in a certain sense. By applying thisresult to Hadwiger’s Covering Problem for certain C -ball convex sets, anddiametrically maximal sets in a Minkowski space, we obtain a stability ofupper bounds on covering numbers.
2. Spindle and ball convex hull, convexity structures
We use the standard notation bd S , int S , relint S , aff S , conv( S ) and card S for the boundary , the interior , the relative interior , the affine hull , the (linear) convex hull and the cardinality of a set S in R n . For two points p, q ∈ R n , [ p, q ]denotes the closed segment connecting p and q . The vectors e , e , . . . , e n ∈ R n denote the standard orthonormal basis of the space, and for a point x ∈ R n , the coordinates with respect to this basis are x = ( x , x , . . . , x n ).The Euclidean norm of p ∈ R n is denoted by | p | . As usual, αA + βB denotesthe Minkowski combination of sets A, B ⊂ R n with coefficients α, β ∈ R (cf.[35]). By an n -polytope we mean an n -dimensional convex body which is the(linear) convex hull of finitely many points. Definition 2.1.
Let C ⊂ R n be a convex body, X ⊆ R n a nonempty set and r ≥ . Then we set B + C ( X, r ) = (cid:92) v ∈ X ( rC + v ) and B − C ( X, r ) = (cid:92) v ∈ X ( − rC + v ) Furthermore, we set B + C ( ∅ , r ) = B − C ( ∅ , r ) = R n . When r is omitted, it is one: B + C ( X ) = B + C ( X, , B − C ( X ) = B − C ( X, . When C = − C we may omit the + / − signs. Note that by Definition 1.2, we have[ p, q ] C = B + C B − C ( { p, q } ) . In the paper we use the following two fundamental concepts.
Definition 2.2.
The spindle convex hull of a set A with respect to C (in short, C -spindle convex hull ), denoted by conv sC ( A ) , is the intersection of all setsthat contain A and are spindle convex with respect to C . The ball convex hullof A with respect to C (or C -ball convex hull ), denoted by conv bC ( A ) , is theintersection of all C -ball convex sets that contain A . We remark that conv bC ( A ) is the intersection of those translates of C that contain C , or in other words, conv bC ( A ) = B + C B − C ( A ).Next, we study the notions of ball and spindle convexity in the contextof the theory of abstract convexity. Zsolt L´angi, M´arton Nasz´odi and Istv´an Talata Definition 2.3. A convexity space is a set X together with a collection G ⊆P ( X ) of subsets of X that satisfy i. ∅ , X ∈ G , and ii. G is closed under arbitrary intersection. Such a collection G of subsets of X is called a convexity structure on X .If a third condition(iii) G is closed under the union of any increasing chains (with respectto inclusion)also holds, then we call the pair ( X, G ) an aligned space (and G an alignedspace structure ). This is the terminology used, for example, by Sierksma [38]and by Kay and Womble [16].We note that in the literature (cf. van de Vel [41]), if G satisfies (i)and (ii), but does not necessarily satisfy (iii), then it is often referred to asa Moore family, or a closure system. On the other hand, ‘convexity space’frequently refers to what we call an aligned space (see [17]). Other terms usedfor an aligned space in the literature are ’domain finite convexity space’ and’algebraic closure system’.The convex hull operation associated with a convexity space ( X, G ) is:conv G ( A ) = ∩{ G ∈ G : A ⊆ G } for any A ⊆ X . The roughest convexity(resp. aligned space) structure G which contains a given family S ⊆ P ( X )is the convexity (resp. aligned space) structure generated by S . This is theintersection of all convexity (resp. aligned space) structures which contain S .For a convex body C ⊆ R n , we denote the family of C -ball convexsets by B C . Clearly, ( R n , B C ) is a convexity space. We note that the familyof C -spindle convex sets is an aligned space structure, while the family ofclosed C -spindle convex sets is a convexity structure. Furthermore, the spaceof C -spindle convex bodies is a geometrical aligned space (an aligned space( X, G ) is called geometrical if A = (cid:83) { conv G ( F ) | F ⊆ A, card( F ) ≤ } forevery A ∈ G , see [17]).Clearly, B C is the convexity space generated by the translates of C .Let T C denote the aligned space structure generated by these translates.In Theorem 1, we compare their corresponding hull operations conv bC andconv T C . For the proof we need the following result of Sierksma [38]. Lemma 2.4 (Theorem 7 in [38] ). Let ( X, S ) be a convexity space, and let G be the aligned space structure generated by S . If A ⊆ X , then conv G ( A ) = ∞ (cid:91) k =0 (cid:20) ∪ { conv S ( F ) : F ⊆ A, card( F ) ≤ k } (cid:21) . Theorem 1.
Let C be a convex body in R n , and let A ⊆ R n . Assume that dim conv bC ( A ) = n . Then cl (conv T C ( A )) = conv bC ( A ) , (1) that is, cl (conv T C ( A )) is the intersection of all translates of C that contain A . all and Spindle Convexity with respect to a Convex Body 5 Proof.
Clearly, cl (conv T C ( A )) ⊆ B + C B − C ( A ). To prove the reverse contain-ment, assume that x ∈ int B + C B − C ( A ). By Lemma 2.4, it is sufficient to showthat x ∈ B + C B − C ( F ) for some finite F ⊆ A .Since − int C + x ⊇ B − C ( A ) (see Remark 5.5), we have ( − bd C + x ) ∩ B − C ( A ) = ∅ . From the compactness of − bd C + x , it follows that there is afinite subset F of A with ( − bd C + x ) ∩ B − C ( F ) = ∅ . Thus, x ∈ B + C B − C ( F ). (cid:3) Clearly, if any C -spindle is n -dimensional, then so is the C -ball convexhull of any set containing more than one point. This leads to the followingobservation. Remark 2.5. If C is a strictly convex body in R n , then cl(conv T C ( A )) =conv bC ( A ) for any set A ⊂ R n . We expect a positive answer to the following question:
Problem 2.6.
Can we drop the condition on the dimension of conv bC ( A ) inTheorem 1?
3. Relationship between spindle and ball convexity, andseparation by translates of a convex body
Clearly, for any convex body C , a C -ball convex set is closed and C -spindleconvex. Thus, for any X and C the C -ball convex hull of X contains its C -spindle convex hull. In [5], it is shown that if C is the Euclidean unit ball,then for closed sets the notions of spindle and ball convexity coincide. Nowwe show that it is not so for every convex body C . Example 3.1.
We describe a -dimensional convex body C and a set H ⊆ R for which H is C -spindle convex but it is not C -ball convex. Let T ⊂ R bea regular triangle in the x = 0 plane, with the origin as its centroid (cf.Figure 1). Let C = conv (cid:0) ( T + e ) ∪ ( − T − e ) (cid:1) . Let H be the intersection of C with the plane with the equation x = 0 . Note that H is a regular hexagon: H = ( T − T ) / . Figure 1.
Example 3.1 Zsolt L´angi, M´arton Nasz´odi and Istv´an Talata
We show that H is C -spindle convex. Note that H and T are of constantwidth two in the two-dimensional norm defined by H . Thus, for any p, q ∈ H ,there is a chord of T , parallel to [ p, q ] , that is not shorter than [ p, q ] . Fromthis, it follows that there is a translate T + z of T that contains p, q . Similarly,there is a set − T + z containing p and q . Denote by K n the collection of all n -dimensional convex bodies. We observe that [ p, q ] C ⊆ C ∩ ( C + z − e ) ∩ ( C + z + e ) ⊆ H , which yields the desired statement.We have that conv sC ( H ) = H . However, clearly, there is only one trans-late of C containing H , namely C . Thus, H = conv sC ( H ) (cid:54) = conv bC ( H ) = C . We introduce the following notions. Note the order of X and Y in Def-inition 3.2. Definition 3.2.
Let C ⊂ R n be a convex body, and let X, Y ⊂ R n . We say thata translate C + x of C separates X from Y if X ⊂ C + x , and int( C + x ) ∩ Y = ∅ . If X ⊂ int( C + x ) and ( C + x ) ∩ Y = ∅ , then we say that C + x strictlyseparates X from Y . In [5], it is proved that if C = B n is the Euclidean ball, then any C -spindle convex set is separated from any non-overlapping C -spindle convexset by a translate of C . By Example 3.1, not all convex bodies have thisproperty (there, H is not separated from any singleton subset of C ). Thus,we introduce the following notions. Definition 3.3.
Let C ⊂ R n be a convex body, and let K ⊂ R n be a C -spindleconvex set. We say that K satisfies Property (p), (s) or (h) with respect to C , if • for every point p / ∈ K , there is a translate of C that separates K from p (Property (p)), • for every C -spindle convex set K (cid:48) that does not overlap with K there isa translate of C that separates K from K (cid:48) (Property (s)), • for every hyperplane H with H ∩ int K = ∅ , there is a translate of C that separates K from H (Property (h)). It is not difficult to show that Property (h) yields (s), which in turnyields (p).
Remark 3.4.
A closed set K satisfies (p) if, and only if, K is C -ball convex. Inparticular, for closed sets the notions of spindle convexity and ball convexitywith respect to a convex body C coincide if, and only if, every closed C -spindleconvex set satisfies (p). Remark 3.5.
For a smooth convex body C , (s) and (h) are equivalent. Recall that in Example 3.1, H is C -spindle convex but not C -ball convex.We note that C may be replaced by a smooth C (cid:48) such that H is C (cid:48) -spindleconvex but not C (cid:48) -ball convex. Simply, apply the following theorem for theconvex body C of Example 3.1 and any smooth and strictly convex body D .Then it follows that H of Example 3.1 is C (cid:48) -spindle convex for C (cid:48) = C + D ,but it is easy to see that H is not C (cid:48) -ball convex.all and Spindle Convexity with respect to a Convex Body 7 Theorem 2.
Let
C, D be convex bodies in R n and let S ⊆ R n . If S is C -ballconvex, then S is ( C + D ) -ball convex. Similarly, if S is C -spindle convex,then S is ( C + D ) -spindle convex. We need the following standard lemma, for a proof see Lemma 3.1.8. in[35].
Lemma 3.6. If C, D are convex bodies in R n , then C = (cid:84) x ∈ D ( C + D − x ) .Proof of Theorem 2. Observe that by Lemma 3.6 we have that conv bC + D ( S ) ⊆ conv bC ( S ), and conv sC + D ( S ) ⊆ conv sC ( S ), for any S ⊆ R n . These readily im-ply the statement of the theorem concerning ball convexity. The statementconcerning spindle convexity follows from the fact that for any two points p, q ∈ S we have [ p, q ] C + D = conv bC + D ( { p, q } ) ⊆ conv bC ( { p, q } ) = [ p, q ] C . (cid:3) A frequently used special case of Theorem 2 is the following.
Corollary 3.7.
Let C be a convex body in R n , let S ⊆ R n , and let < r < bearbitrary. If S is C -ball convex, then rS is C -ball convex. In particular, rC isa C -ball convex set. Similarly, if S is C -spindle convex, then rS is C -spindleconvex. In particular, rC is C -spindle convex.Proof. We apply Theorem 2 to rC and (1 − r ) C . (cid:3) For n ≥
3, the analogous implication of Theorem 2 is not true for S + D in general, so ball convexity is not preserved in general by adding a convexbody D to both a C -ball convex set S and to C . The same holds for spindleconvexity. We show both in the following example. Example 3.8.
We describe convex bodies
C, D ⊂ R n and a set S ⊂ R n , forany n ≥ , such that S is C -ball convex (and thus S is C -spindle convex),and S + D is not ( C + D ) -spindle convex. We note that all sets C, D, S ⊆ R n will be centrally symmetric.Let C = conv([ − , n ∪ {± (1 + ε ) e i } n − i =1 ) , where < ε < . Let D = rB n , that is, D is a Euclidean ball of radius r for some r > , centered atthe origin. Choose S = [ e n , − e n ] . Then S = conv bC ( S ) = conv sC ( S ) since S = (cid:16) C + (cid:80) n − i =1 e i (cid:17) ∩ (cid:16) C − (cid:80) n − i =1 e i (cid:17) .On the other hand, ( S + D ) ∩ { x n = 0 } = rB n − × { } , and thus ± re i ∈ bd( S + D ) for any ≤ i ≤ n − , while we have ± re i ∈ int (cid:18) conv sC + D (cid:16) (1 + r ) e n ∪ (cid:0) − (1 + r ) e n (cid:1)(cid:17)(cid:19) ⊆ int (cid:0) conv sC + D ( S + D ) (cid:1) , for any ≤ i ≤ n − , so ( S + D ) (cid:40) conv sC + D ( S + D ) . In the following, we show that Example 3.1 is not a ‘rare phenomenon’.
Theorem 3.
Let n ≥ . The family of those smooth n -dimensional convexbodies for which the associated ball and spindle convexity do not coincide,forms an everywhere dense set in the metric space of the n -dimensional con-vex bodies, equipped with the Hausdorff metric. Zsolt L´angi, M´arton Nasz´odi and Istv´an Talata
Proof.
Let C be any convex body. Note that there are two distinct points x, y of C with a hyperplane F through the origin such that the parallelhyperplanes F + x and F + y support C and their intersection with C is { x } and { y } , respectively.Let T be a small ( n − F with the originas its centroid. Let C (cid:48) = conv( C ∪ ( T + x ) ∪ ( − T + y )) and let H = ( T − T ) / H is an ( n − C (cid:48) -spindle convex set, whereas conv bC (cid:48) ( H ) is n -dimensional.To construct a smooth body with respect to which ball and spindleconvexity do not coincide, we take H and ¯ C = C (cid:48) + ρB n . If ρ > C is close to C . (cid:3) Now we present an example of C and a closed convex set H where H satisfies Property (p), i.e., it is C -ball convex, but does not satisfy (h). Figure 2.
Example 3.9
Example 3.9.
Let T be the Euclidean unit disk centered at the origin in theplane U = { x = 0 } in R , and consider the points p = e and q = − e .Let T (cid:48) ⊂ T + e be a smooth plane convex body in the U + e plane withthe following properties: T (cid:48) is symmetric about e , and the chord [ p (cid:48) , q (cid:48) ] of T (cid:48) connecting its supporting lines parallel to the x -axis is parallel to and slightlylonger than [ p, q ] . We choose our notation in a way that T (cid:48) + p − p (cid:48) and T areon the same side of the line L defined by the equations x = 1 , x = 0 . Let p ∗ be the reflection of p (cid:48) in the plane U , and set C = conv( T ∪ T (cid:48) ∪ ( − T (cid:48) )) .Figure 2 shows the x = 0 and x = 1 sections of C , viewing it from thetop. Let H = ( C + p − p (cid:48) ) ∩ ( C + p − p ∗ ) and observe that H = T (cid:48) + p − p (cid:48) is a C -ball convex set contained in U . Consider the plane with the equation x = 1 , and note that it supports H at p . all and Spindle Convexity with respect to a Convex Body 9 If a translate C + v of C separates H from the plane x = 1 then p ∈ C + v , and x = 1 is a support plane of C + v . It follows that v = 0 .However, C (cid:43) H because | p (cid:48) − q (cid:48) | > | p − q | . A suitable modification of this example yields a smooth convex body C arbitrarily close to the Euclidean unit ball such that some C -ball convex sets(which thus satisfy (p)) do not satisfy (h). Hence, by Remark 3.5, there arespindle convex sets that satisfy (p) and do not satisfy (s) for some convexbody C .We propose the following problems. Problem 3.10.
Is there a spindle convex set K that satisfies (h) and does notsatisfy (s) with respect to some convex body C ? Problem 3.11.
Is there a convex body C such that every C -spindle convex set K satisfies (p) (respectively (s)), but at least one of them does not satisfy (s)(respectively (h))? We conclude this section by finding some special classes of convex bodiessuch that any spindle convex set with respect to any of them satisfies Property(h). Thus, if a set is closed and spindle convex with respect to any of them,then it is ball convex with respect to the same convex body. First, we recalla standard definition.
Definition 3.12.
The central symmetral ( C − C ) of a convex body C ⊂ R n defines a norm on R n , called the relative norm of C (cf. [10] ). We recall thatin this norm, C is a convex body of constant width two. For points p, q ∈ R n ,we call the distance between p and q , measured in the norm relative to C , the C -distance of p and q (cf. Lassak [23] ). For a set X , we denote the diameterof X measured in C -distance by diam C X . If C is the Euclidean ball, we maywrite simply diam X . Theorem 4.
Let C ⊂ R be a plane convex body. Then any K ⊂ R n C -spindleconvex set satisfies (h).Proof. Clearly, we may assume that K is closed. We show the assertion forthe case that K is a plane convex body. We leave it to the reader to verify itin the cases that K is not bounded or has an empty interior.Let K be a C -spindle convex body, and L be any line supporting K .Since translating C does not change whether K satisfies (h) or not, we mayassume that L supports also C . Let L (cid:48) be the other supporting line of C parallel to L .First, consider the case that C ∩ L is a singleton { x } . Note that as K is C -spindle convex, we have that K ∩ L is also a singleton, since otherwisethe C -spindle of the endpoints of K ∩ L has a point on the side of L notcontaining K . Without loss of generality, we may assume that K ∩ L = { x } .Assume that K (cid:54)⊆ C , and consider a point y ∈ K \ C . Observe thatthere is a translate of C containing x and y if, and only if their C -distanceis at most two. Thus, y is contained in the closed unbounded strip between0 Zsolt L´angi, M´arton Nasz´odi and Istv´an Talata L and L (cid:48) . Note that if we sweep through C by a family of parallel lines,the C -length of the intersecting segment strictly increases while its lengthreaches two, then it may stay two for a while, then it strictly decreases untilit reaches the other supporting line of C . Consider the chords of C that areparallel to and not shorter than [ x, y ], and observe that, since y / ∈ C , theyall are on the side of the line, passing through x and y , that contains L (cid:48) ∩ C (cf. Figure 3). From this, it follows that the intersection of all the translatesof C containing x and y (or, in other words, the C -spindle of x and y ) has apoint on the side of L not containing C . Since K is C -spindle convex and L supports K at x , we arrived at a contradiction. Figure 3.
When C ∩ L = { x } Figure 4.
When C ∩ L = [ p, q ]Now consider the case that C ∩ L = [ p, q ] with p (cid:54) = q . Let K ∩ L = [ s, t ].Then | s − t | ≤ | p − q | , and we may assume that [ s, t ] ⊂ [ p, q ] and that t ∈ [ s, q ].If there is a point v ∈ K not contained in the closed strip bounded by L and L (cid:48) , then the C -distance of v and s is greater than two; a contradiction. Set C = t − q + C , C = s − p + C and u = q − p − ( s − t ), which yields that C = u + C .Assume that there is no translate of C , with supporting line L , thatcontains K . Then, C λ \ K (cid:54) = ∅ for every λ ∈ [0 , C λ = λu + C . Let H denote the closed strip bounded by L and L (cid:48) . For simplicity, we regard theconnected component of H \ C λ , not containing q as the one below C λ , andthe other one as the one above C λ . It is easy to see that if there is no λ with K ⊂ C λ , then there is a value of λ such that both components of H \ C λ contain a point of K . Let λ be such a value, and let x, y ∈ K be points suchthat x is in the component below C λ and y is in the component above C λ (cf. Figure 4).Consider the segment [ x, y ] and note that if their C -distance is greaterthan two, then [ x, y ] C = R . Thus, all the chords of C λ that are parallelto and not shorter than [ x, y ] are on the same side of the line containingall and Spindle Convexity with respect to a Convex Body 11[ x, y ]. Similarly like in the previous case, it follows that [ x, y ] C (cid:54)⊂ H , whichcontradicts our assumption that K (cid:54)⊆ H . (cid:3) Corollary 3.13.
For a plane convex body C ⊂ R , any closed C -spindle convexset is C -ball convex. Proposition 3.14.
Let C ⊂ R k and C ⊂ R m be convex bodies, and considera set S ⊂ R k ⊕ R m = R k + m . Let proj and proj be the orthogonal projectionsof R k ⊕ R m on the first and second factor, respectively. Then, conv bC × C ( A ) = conv bC (proj A ) × conv bC (proj A ) ;conv sC × C ( A ) = conv sC (proj A ) × conv sC (proj A ) . Proof.
Note that for any x ∈ R k + m , x + ( C × C ) = (proj x + C ) × (proj x + C ). Thus, A ⊆ x + ( C × C ) is equivalent to proj A ⊆ (proj x + C ) and proj A ⊆ (proj x + C ). This immediately yields the equality ofthe ball convex hulls. It follows that for any points p, q ∈ R k + m , we have[ p, q ] C × C = [proj p, proj q ] C × [proj p, proj q ] C . Hence, the right-handside in the second equality contains the left-hand side. Now consider any ( C × C )-spindle convex set K . We show that proj K × proj K ⊆ K . Considerpoints p ∈ proj K and q ∈ proj K . We need to show that ( p , q ) ∈ K .Note that ( p , p ) ∈ K and ( q , q ) ∈ K for some p ∈ R m and q ∈ R k .Thus, by applying the first equality for ( C × C )-spindles, we obtain that( p , q ) ∈ K , which is what we wanted to prove. (cid:3) Corollary 3.15. If C is an n -dimensional axis-parallel cube, and A ⊆ R n isany set, then both conv bC ( A ) and conv sC ( A ) are either the axis-parallel boxcontaining A and minimal with respect to inclusion, or R n .
4. Arc-distance defined by C In the theory of spindle convexity with respect to the Euclidean disk, thereis a naturally arising associated distance function. This distance is called arc-distance , and is defined for points p, q ∈ R as the Euclidean length ofa shortest unit circle arc connecting the points (cf. [4] and [5]). The aim ofthis section is to generalize this distance for spindle convexity with respectto any origin-symmetric plane convex body.Let C be a planar o -symmetric convex body, that is the unit disk ofa normed plane. We recall that the C -length of a polygonal curve is thesum of the C -distances between the consecutive pairs of points, and thatthe C -length of a curve is the supremum, if it exists, of the C -lengths of thepolygonal curves for which all the vertices are chosen from the curve. If D is aconvex body, then the perimeter of D with respect to C is the C -length of theboundary of D . We denote this quantity by perim C D . It is known that forany plane convex bodies C and D , we have perim C D = perim C ( ( D − D ))(cf. [11]), and 6 ≤ perim C C ≤ Definition 4.1.
Let C be an o -symmetric plane convex body, and let p, q bepoints at C -distance at most two. Then the arc-distance ρ C ( p, q ) of p, q withrespect to C is the minimum of the C -lengths of the arcs, with endpoints p and q , that are contained in bd( y + C ) for some y ∈ R . Definition 4.2.
Let C be an o -symmetric plane convex body, z ∈ R and ≤ ρ ≤ perim C C . Then the arc-distance disk, with respect to C , of center z and radius ρ is the set ∆ C ( z, ρ ) = { w ∈ R : ρ C ( z, w ) ≤ ρ } . Furthermore, we set ∆ C ( ρ ) = ∆ C ( o, ρ ) . Clearly, for any C , arc-distance disks of the same radius are translatesof each other, but those of different radii are not necessarily even similar. Wenote that if C is a Euclidean disk, then its arc-distance disks are Euclideandisks. Theorem 5.
For any o -symmetric plane convex body C and ≤ ρ ≤ perim C C ,the arc-distance disk ∆ C ( ρ ) is convex.Proof. Note that by compactness arguments, it is sufficient to prove the as-sertion for the case that C is a convex polygon with, say, m vertices and forvalues of ρ such that 0 ≤ ρ ≤ perim C C is not equal to the sum of somesides of C . In this case no chord of C connecting two vertices determines anarc of length ρ .Let us move a point p ( t ) around on bd C at a constant speed measuredin C -distance, and consider the point q ( t ) such that ρ C ( p ( t ) , q ( t )) = ρ . Notethat on any side of C , the points p ( t ) and q ( t ) move at a constant speedalso in the Euclidean metric, and their Euclidean speed is proportional tothe lengths of the longest chords of C parallel to the corresponding edges of C . Thus, the vector q ( t ) − p ( t ) is a linear function of t if p ( t ) and q ( t ) areon different edges, and a constant if they are on the same edge, which yieldsthat Q = bd ∆ C ( ρ ) is a (starlike) polygonal curve with at most 2 m vertices. Figure 5.
Polygon C in the proof of Theorem 5all and Spindle Convexity with respect to a Convex Body 13We show that Q is convex. Consider four points: a vertex of Q , onepoint on each edge meeting at that vertex and the origin. Since Q is clearlystarlike with respect to the origin, it is sufficient to prove that for any choiceof the vertex (and the points on the edges), these four points are in convexposition.Any vertex of Q is of the form q − p , where q is a vertex of C , and p isa point in the relative interior of some edge of C .Consider the sufficiently small vectors v , v and v whose Euclideanlengths are proportional to the Euclidean lengths of the longest chords of C parallel to them, and such that p − v , p , p + v are on the same edge of C , and q + v and q + v are on the two consecutive edges of C meetingat q . We choose our notation in a way that the line passing through [ p, q ]separates p − v and q + v from p + v and q + v , and that v , − v and v are in counterclockwise order (cf. Figure 5). Note that as C is convex, thesegment [ q + v , q + v ] intersects [ p, q ].Now, to show that Q is convex, we need to show that the points o , q + v − ( p − v ), q − p and q + v − ( p + v ) are in convex position. Adding p toeach point, we are considering the quadrilateral with vertices p , q + v + v , q and q + v − v . It is clearly convex, by the convexity of C . (cid:3) We note that if C is not o -symmetric, we may define a non-symmetricarc-distance, for which we may prove the assertion of Theorem 5 using asimilar technique. Remark 4.3. If C is smooth, then ∆ C ( ρ ) is smooth for any positive value of ρ .Proof. Consider the C -arc-length parametrization Γ : [0 , α ] → R of bd C .Then bd ∆ C ( ρ ) is the graph of the curve τ (cid:55)→ Γ( τ + ρ ) − Γ( τ ). If Γ is differ-entiable, then so is this function. (cid:3) Our next corollary follows from Theorem 5 and the fact that ∆ C ( ρ ) ⊂ ∆ C ( ρ ) for any ρ < ρ . Corollary 4.4.
For any x, y, z ∈ R and an o -symmetric plane convex body C , the function τ (cid:55)→ ρ C ( x, y + τ z ) is quasiconvex on its domain; that is:it consists of a strictly decreasing, possibly a constant and then a strictlyincreasing interval. We prove the following version of the triangle inequality for arc-distance,which, for the Euclidean case, appeared first as Lemma 1 in [4].
Theorem 6.
Let C be an o -symmetric plane convex body, and let x, y, z ∈ R be points such that each pair has a C -arc-distance. (1) If y ∈ int[ x, z ] C , then ρ C ( x, y ) + ρ C ( y, z ) ≤ ρ C ( x, z ) , (2) if y ∈ bd[ x, z ] C , then ρ C ( x, y ) + ρ C ( y, z ) = ρ C ( x, z ) , and (3) if y / ∈ [ x, z ] C and C is smooth, then ρ C ( x, y ) + ρ C ( y, z ) ≥ ρ C ( x, z ) . Proof.
The assertion of (2) is trivial. Assume that y ∈ int[ x, z ] C . Clearly,there is a line L containing y such that L ∩ bd[ x, z ] C = { a, b } with ρ C ( x, a ) = ρ C ( x, b ) and ρ C ( z, a ) = ρ C ( z, b ). Thus, the assertion in this case is a conse-quence of Corollary 4.4.Assume that C is smooth and y / ∈ [ x, z ] C . If ρ C ( x, y ) ≥ ρ C ( x, z ), thenthere is nothing to prove, and thus, we may assume that y ∈ int ∆ C ( x, ρ C ( x, z )).Similarly, we may assume that y ∈ int ∆ C ( z, ρ C ( x, z )). For any 0 < τ <ρ C ( x, z ), let L ( τ ) denote the line for which L ( τ ) ∩ bd[ x, z ] C = { c, d } with ρ C ( x, c ) = ρ C ( x, d ) = τ , and let R be the region swept through by the lines L ( τ ), 0 < τ < ρ C ( x, z ). Since C is smooth, the limit L + of L ( τ ) as τ ap-proaches ρ ( x, z ) is the supporting line of ∆ C ( x, ρ ( x, z )) at z . Similarly, if τ →
0, the limit L − of L ( τ ) is the supporting line of ∆ C ( z, ρ C ( x, z )) at x .Since ∆ C ( x, ρ C ( x, z )) ∩ ∆ C ( z, ρ C ( x, z )) lies between the parallel lines L + and L − , and this open unbounded strip is clearly contained in R , y ∈ L ( τ ) forsome value of τ , and (3) follows from Corollary 4.4. (cid:3) Example 4.5.
Let C be the unit ball of the l ∞ norm in R . Then ∆( ρ ) = (cid:26) { ( x , x ) ∈ R : | x | + | x | ≤ ρ } , if < ρ ≤ , { ( x , x ) ∈ R : | x | + | x | ≤ ρ, | x | ≤ , | y | ≤ } , if < ρ ≤ . In this example, for any two points x, z with ρ C ( x, z ) ≤
2, we have ρ C ( x, y ) + ρ C ( y, z ) = ρ ( x, z ) for any y ∈ [ x, z ] C . Furthermore, if we replacethe corners of C with small circle arcs, then the boundary of the arc-distanceballs of the obtained body consists ’almost only’ of segments that are parallelto the segments in bd ∆( ρ ). Thus, we may create a smooth convex body andpoints x, y, z with y / ∈ int[ x, z ] C such that ρ C ( x, y ) + ρ C ( y, z ) = ρ C ( x, z ).We propose the following questions. Problem 4.6.
Can we drop the smoothness condition in part (3) of Theo-rem 6?
Problem 4.7.
Prove or disprove that if C is strictly convex, then ∆ C ( ρ ) isstrictly convex for any < ρ ≤ perim C C . Problem 4.8.
Prove or disprove that if C is strictly convex, then the inequal-ities in (1) and (3) of Theorem 6 are strict.
5. Carath´eodory numbers
Now we recall the notion of the Carath´eodory number of a convexity space(cf. [22], [33] and [37]).
Definition 5.1.
Let ( X, G ) be a convexity space (for the definition, see Sec-tion 2). The Carath´eodory number Car G of G is the smallest positive integer k such that for any V ⊆ X and p ∈ conv G ( V ) there is a set W ⊆ V with card W ≤ k and with p ∈ conv G ( W ) . If no such positive integer exists, we set Car G = ∞ . all and Spindle Convexity with respect to a Convex Body 15 Definition 5.2.
Let C ⊂ R n be a convex body, and let G (respectively G ) bethe family of closed C -spindle convex sets (respectively, the C -ball convex sets)in R n . Then we call Car G (respectively Car G ) the spindle Carath´eodorynumber (respectively, ball Carath´eodory number ) of C , and denote it by Car s C (respectively, by Car b C ). These numbers were determined in [5] for the Euclidean ball B n as C . Theorem 7.
Let C ⊂ R be a plane convex body. If C is a parallelogram, thenboth Carath´eodory numbers of C are two, otherwise both are three.Proof. By Theorem 4, for any C ⊂ R , a closed set is C -spindle convex if,and only if it is C -ball convex. Thus, the two Carath´eodory numbers of C are equal.Let X ⊂ R be any closed set. The C -ball convex hull of X is theintersection of all the translates of C that contain X . If X is a singleton, theassertion immediately follows, and thus we assume that card X >
1. If notranslate of C contains X , then, by Helly’s theorem, there are at most threepoints of X that are not contained in any translate of C .Assume that there is a translate of C containing X . Consider a point p ∈ bd conv sC ( X ). If p ∈ X , we are done. Assume that p / ∈ X . We leaveit to the reader to show that there is a translate u + C with the propertythat bd( u + C ) contains p and two distinct points z , z ∈ X , such that the C -distance of z and z is 2, or, if the C -distance of z and z is less thantwo, then the connected component of ( u + C ) \ [ z , z ] containing p does notcontain points at C -distance two. Without loss of generality, we may assumethat the open arc in bd( u + C ) with endpoints z and z that contains p isdisjoint from X .If the C -distance of z and z is less than two, then, clearly, p is containedin any translate of C that contains z and z , or, in other words, p ∈ [ z , z ] C .If the C -distance of z and z is equal to two, there are two parallel lines L and L that support u + C at z and z , respectively. Since y ∈ conv sC ( X ),we have for i = 1 , z i is the endpoint of the segment L i ∩ ( u + C ) closerto p . Thus, p ∈ [ z , z ] C .Now consider a point p ∈ int conv sC ( X ). Let v ∈ X be arbitrary. Choosea point z ∈ bd conv sC ( X ) such that p ∈ [ v, z ]. Then, by the previous para-graph, there are points z , z ∈ X such that z ∈ [ z , z ] C , and clearly, p ∈ conv sC ( { v, z , z } ).Finally, assume that C is not a parallelogram. Note that in that casethere are three smooth points in bd C such that the unique lines supporting C at them are the sidelines of a triangle containing C . Let X be the vertexset of this triangle. Observe that for a sufficiently large λ >
0, the centroid c of X is not contained in the ( λC )-spindles determined by any two points of X . Since c ∈ conv( X ) ⊆ conv sλC ( X ) for any value of λ , we have that bothCarath´eodory numbers of C are three. The observation that they are two if C is a parallelogram follows from the next theorem. (cid:3) Proposition 5.3. If C is an n -dimensional parallelotope, then Car s C = Car b C = n .Proof. Since both Carath´eodory numbers are affine invariant quantities, wemay assume that C is the unit ball of the l ∞ norm. By Corollary 3.15, thetwo Carath´eodory numbers of C are equal. Let X be a closed set in R n . Thenconv sC ( X ) is either R n or the minimal volume axis-parallel box that contains X . In the first case it is easy to see that there are two points of X that arenot contained in any translate of C , from which the assertion readily follows.Assume that conv sC ( X ) is the minimal volume axis-parallel box thatcontains X , and consider a point y ∈ conv sC ( X ). Then, by a theorem of Lay[25], there is a subset X (cid:48) ⊆ X with card X (cid:48) ≤ n such that the minimal volumeaxis-parallel box containing X (cid:48) contains y . Thus, we obtain y ∈ conv sC ( X (cid:48) )and hence Car s C ≤ n .On the other hand, let X be the set of those n vertices of C which areconnected by an edge of C with a given vertex v of C . Then conv sC ( X ) = C .Let y be the vertex of C opposite of v , and observe that, removing any pointof X , the C -ball convex hull of the remaining points is a facet of C notcontaining y . Thus, Car s C ≥ n . (cid:3) Similarly to Corollary 3.15 and Proposition 5.3, we can prove the fol-lowing.
Proposition 5.4.
Let C ⊂ R n be a simplex. (i) For any closed set X ⊂ R n , both conv bC ( X ) and conv sC ( X ) are either R n , or the smallest positive homothetic copy of C that contains C . (ii) Car s C = Car b C = n + 1 . We apply the following remark several times in this section.
Remark 5.5.
For any set H ⊂ R n , a point p ∈ R n , and a convex body C ⊂ R n ,we have that p ∈ B − C ( H ) if, and only if, C + p ⊇ H . Similarly, p ∈ B + C ( H ) if, and only if, − C + p ⊇ H . Proposition 5.6.
Let C ⊂ R n be an n -polytope with k facets. Then the ballCarath´eodory number of C is at most kn .Proof. Let X ⊂ R n and p ∈ conv bC ( X ) be given. We need to find a subset Y of X of cardinality kn such that p ∈ conv bC ( Y ). By Remark 5.5, we havethat C ⊇ B − C ( X ), that is, ( R n \ C ) ∩ (cid:84) v ∈ X ( − C + v ) = ∅ . Since R n \ C is theunion of k convex sets, the statement follows from Helly’s theorem. (cid:3) The following construction, similar to Example 4 in [31], shows that in R there are convex bodies with arbitrarily large ball Carath´eodory numbers. Theorem 8.
Let k ∈ Z + be given. Then there is an o -symmetric convex body C ⊂ R and a set X ⊂ R such that o ∈ conv bC ( X ) , but for any Y ⊂ X , card Y < k we have that o / ∈ conv bC ( Y ) . all and Spindle Convexity with respect to a Convex Body 17 Proof.
We may assume that k is even. Consider the paraboloid P in R defined by x = x + x . We choose k points on the parabola P = P ∩{ x = 0 } , and number these points according to the order in which theylie on the parabola: U = { u , u , . . . , u k } ⊂ P . Plane sections of P parallelto the x x -plane are translates of this parabola. Let P i = P ∩ { x = i } for i ∈ I , where I = (cid:8) − k , − k + 1 , . . . , − , − , , , . . . , k (cid:9) . Now, for each i ∈ I , there is a unique translation vector t i such that P = P i + t i . Let U i = ( U \ { u i } ) − t i ⊂ P i . Let h > x coordinateof the points in any U i . Finally, consider the (bounded) arc of P that liesin the half-space { x ≤ h } . Delete from this arc very small open arcs aroundeach point of U , and call the remaining part of P (the union of k + 1 closedbounded arcs) U . We define C as the following o -symmetric convex body: C = conv (cid:34)(cid:32) k (cid:91) i =0 U i − (0 , , h ) (cid:33) (cid:91) − (cid:32) k (cid:91) i =0 U i − (0 , , h ) (cid:33)(cid:35) . Let X = { t i : i ∈ I } . Now, B C ( X ) ⊆ B C (cid:110) t − k , t k (cid:111) , and the latteris contained in the x = 0 plane. Moreover, B C ( X ) is contained in the pla-nar region conv( P − (0 , , h )) ∩ − conv( P − (0 , , h )). If the open arcs in thedefinition of U are sufficiently small then a little more is true: B C ( X ) is con-tained in the planar region conv( U − (0 , , h )) ∩ − conv( U − (0 , , h )). Thus, B C ( X ) ⊂ C . It follows, by Remark 5.5, that o ∈ B C B C ( X ) = conv bC ( X ). Onthe other hand, for any i ∈ I , we have that u i ∈ B C ( X \{ t i } ), and hence, C (cid:54)⊃ B C ( X \{ t i } ). It follows (again by Remark 5.5) that o (cid:54)∈ conv bC ( X \ { t i } ). (cid:3) This example may be modified in several ways. First, we may generalizeit for R n with n >
3, by replacing C with C × [ − , n − and leaving X un-changed. Second, by “smoothening” C , we may obtain a smooth and strictlyconvex o -symmetric body C in R n with an arbitrarily large ball Carath´eodorynumber. Third, we may replace U by a sufficiently dense finite subset of U ,and thus obtain a polytope as C . Finally, the following modification of theexample yields an o -symmetric convex body in R whose ball Carath´eodorynumber is infinity: In the construction, replace the finitely many P i s by planarsections of P of the form P i = P ∩ { x = a i } where a , a , . . . is a sequenceof real numbers in ( − , U similarly: let u , u , . . . be a bounded sequence of points on P , which does not contain anyof its accumulation points. Problem 5.7.
For n ≥ , find the minima of the ball/spindle Carath´eodorynumbers of the n -dimensional convex bodies, and if it exists, find the maxi-mum of their spindle Carath´eodory numbers. Problem 5.8.
For n ≥ , prove or disprove the existence of a convex body C ⊂ R n such that its two Carath´eodory numbers are different. Problem 5.9.
It is known (cf. [5] ) that both Carath´eodory numbers of the n -dimensional Euclidean ball are n + 1 . Prove or disprove that this holds also in a small neighborhood of the Euclidean ball. If the answer is negative, isthe set of ball (resp., spindle) Caratheodory numbers bounded from above ina neighborhood of the Euclidean ball?
6. Finitely generated C -ball convex sets Clearly, every C -spindle is C -ball convex, but the converse is not true ingeneral. However, there are convex bodies C for which every C -ball convexset is a C -spindle, such as the simplices and rectangular boxes of R n , seeCorollary 3.15 and Proposition 5.4. In this section, we examine a more generalproblem: We investigate those convex bodies C that have the property thatevery C -ball convex set is the C -ball convex hull of finitely many points. Definition 6.1.
If every C -ball convex set is the C -ball convex hull of at most k points, for some fixed k ≥ , then we say that every C -ball convex set is k -generated . Similarly, if every C -ball convex set is the C -ball convex hullof finitely many points, then we say that every C -ball convex set is finitelygenerated . Note that if every C -ball convex set is k -generated for some convex body C ⊂ R n , where k ≥
2, and P is a C -ball convex set, card( P ) >
1, then P canbe obtained as the C -ball convex hull of exactly k points of R n . Theorem 9. If C is a convex body in R n for which every C -ball convex set is k -generated, then C is an n -polytope with at most kn facets. This theorem is a consequence of the next three lemmas. The first ofthese readily follows by the minimality property of C -ball convex hull as theintersection of translates of C . Lemma 6.2.
Let C be a convex body in R n . Let K ⊆ R n be a bounded set, andassume that K = conv bC ( S ) for some closed set S ⊆ K . Then, a translate C + v contains K and fulfils bd( C + v ) ∩ K (cid:54) = ∅ if and only if bd( C + v ) ∩ S (cid:54) = ∅ . Lemma 6.3. If C ⊂ R n is an n -polytope for which every C -ball convex set is k -generated, then C has at most kn facets. Lemma 6.4.
If every C -ball convex set is finitely generated for a convex body C ⊂ R n , then C is an n -polytope.Proof of Lemma 6.3. We can obtain C as the intersection of finitely manytranslates of C : C = (cid:84) mi =1 ( C + v i ). We perturb the translation vectors v i to obtain another polytope P = (cid:84) mi =1 ( C + w i ) in a way that | v i − w i | < ε fora sufficiently small ε >
0, and P is a simple polytope (that is, every vertexof C is contained in exactly n facets of C ), and each facet of P is containedin the relative interior of a facet of C + w i for some value of i , and the n i sare pairwise distinct. By Lemma 6.2, there are k points of P such that everyfacet of P contains at least one of these points. Since any point is containedin at most n facets of P , we obtain m ≤ kn . (cid:3) all and Spindle Convexity with respect to a Convex Body 19 Proof of Lemma 6.4.
Assume that C ⊂ R n is a convex body which is not apolytope, and let 0 < r <
1. It follows from Theorems 2.2.4 and 2.2.9 of [35]that there is an infinite sequence of pairwise distinct triples T i = ( p i , H i , n i ), i = 1 , , , . . . , such that p i is a smooth boundary point of rC , H i is theunique supporting hyperplane of rC at p i , and n i is the outer normal unitvector of H i with respect to rC , for which n i (cid:54) = n j if i (cid:54) = j . To see thisdirectly, it is also easy to construct such triples applying induction.We choose an infinite subsequence { T i | i ∈ I } of the triples, for whichthere is at most one cluster point of { p i | i ∈ I } and { n i | i ∈ I } , resp., andthat cluster point is not equal to any p i and n i .Let H + i be the half-space determined by H i which contains rC . Let r < r (cid:48) <
1. Let C i be that translate of r (cid:48) C which touches both rC and H i at p i , i ∈ I . Let P = (cid:84) i ∈ I C i . Since there is no cluster point among the points p i and vectors n i ( i ∈ I ), we can define a sequence B i of balls of positive radiisuch that for every i ∈ I , B i is centered at p i , and B i ⊆ int( C j ) for any j (cid:54) = i .Therefore, every C i can be translated by a vector v i towards the direction n i within a sufficiently small, but positive distance such that the translates C (cid:48) i = C i + v i , i ∈ I form P (cid:48) = (cid:84) i ∈ I C (cid:48) i in such a way that H (cid:48) i = H i + v i is the unique supporting hyperplane of P (cid:48) at p (cid:48) i = p i + v i , and at most n hyperplanes have a common point among the hyperplanes H i ( i ∈ I ).Then, since the smaller homothetic copies C i and C (cid:48) i ( i ≥
1) of C are C -ball convex sets by Corollary 3.7, by assumption, P (cid:48) is a C -ball convexhull of finitely many points, so there should be a finite subset of bd( P (cid:48) ) suchthat every face H (cid:48) i ∩ P (cid:48) i contains at least one element of S (otherwise therewould be a translate of C touching P (cid:48) and having a disjoint boundary fromthose points, and by Lemma 6.2, that would contradict the fact that P (cid:48) isa C -ball convex hull of those points). But one point can be contained in atmost n faces among the infinitely many faces H (cid:48) i ∩ P (cid:48) i , so P (cid:48) can not be the C -ball convex hull of finitely many points. (cid:3) Theorem 9 implies the following corollary.
Corollary 6.5.
Let C be a convex body in R n . If every C -ball convex set in R n is a C -spindle, then C is a polytope and it has at most n facets. While Corollary 6.5 is sharp for parallelotopes, we do not know if it isthe case for Theorem 9 for k ≥
3. We ask the following question.
Problem 6.6.
Let k ≥ be arbitrary. Is there an n -polytope C ⊂ R n with kn facets such that every C -ball convex set is k -generated? In the first part of this section, we find an upper bound for the number offacets of those polytopes C for which every C -ball convex set is k -generated.It is also natural to estimate these numbers from below. Now, we consider thefollowing problem: For a fixed integer k ≥
2, what is the maximum number m = m ( n, k ) for which every n -polytope C that has at most m facets, alsohas the property that every C -ball convex set is k -generated? We have thefollowing partial solution for this problem.0 Zsolt L´angi, M´arton Nasz´odi and Istv´an Talata Theorem 10.
Let n ≥ . (1) If ≤ k ≤ n , then there is an n -polytope C ⊂ R n having n + k + 2 facetssuch that not every C -ball convex set is k -generated. (2) If k ≥ , and C ⊂ R n is any n -polytope having at most n + k + 1 facets,then every C -ball convex set is k -generated. The example of a pentagon shows that the assertion in (2) fails for n = k = 2. It is easy to see that (2) also holds for n = 2 and k ≥ Proof.
To prove (1), assume 2 ≤ k ≤ n , and let C ⊂ R n be the n -polytopewhich is obtained from an n -simplex S n by intersecting it with k closed half-spaces near k vertices of S n so that the k new facets are pairwise disjoint.Let 0 < r < T of k points, rC has afacet disjoint from T . Thus, by Lemma 6.2, rC is not the C -ball convex hullof at most k points.Next, we prove (2). Now assume that k ≥ C ⊂ R n is an n -polytopewith n + k + 1 facets. Let P be an arbitrary C -ball convex set of R n , P (cid:54) = R n .Then P is a polytope of at most n + k + 1 facets. We will assume that C is a simple n -polytope. We may do so, since it is easy to see that for every C -ball convex set P there is a sequence { P i } ∞ i =1 of simple n -polytopes suchthat every P i is a C i -ball convex set, and P i → P , C i → C in the Hausdorffmetric, as i → ∞ , where C i ⊂ R n is an n -polytope for every i = 1 , , , . . . .Then any sequence of at most k element subsets which span P i as a C i -ballconvex hull, for i = 1 , , , . . . , has a subsequence whose elements converge toan at most k element subset of P . Clearly, the C -ball convex hull of this setis P . By a similar limit argument, we may further assume that dim P = n , P is a simple polytope, and P has exactly n + k + 1 facets.We need to show that there are k points whose C -ball convex hull is P .By Lemma 6.2, it is sufficient to prove that there are k points of P such thatfor each facet F of C there is a translate C + t of C ( t ∈ R n ) that contains P and for which F + t contains at least one of the k points. With the aboveassumption on the number of facets and dimension of P , it is equivalent tothe existence of a set T of k points such that every facet contains at least oneelement of T .Let v be a vertex of P . Let H be a hyperplane such that v / ∈ H and H is parallel to a supporting hyperplane H (cid:48) of P at v for which H (cid:48) ∩ P = { v } .Let π be the central projection of R n (cid:114) H (cid:48) to H from the point v , that is π ( x ) = aff( v, x ) ∩ H . Then π ( P ) is bounded, in fact, it is an ( n − S n − since P is a simple polytope. Consider the projection of thefacets of P under π . Then the images of those facets which contain v arethe facets of an S n − , and we denote them by F = { F , F , . . . , F n } . Theimages of the remaining k + 1 facets form a tiling of S n − , we denote themby A = { A , A , . . . , A k +1 } . Clearly, every A i is an ( n − V the vertex set of S n − .Obviously, there are k points that span P as the C -ball convex hull ofthose points if and only if either there are k − S n − such thatall and Spindle Convexity with respect to a Convex Body 21every A i contains at least one of them, or there are k points of S n − such thatevery A i and F j contains at least one of them. If there are three elementsof A having a common point, then such a set of k − A and one point from each remaining element of A . So, from now on, we mayassume that there is no common point of three elements of A .Since A contains more than one element, no A i contains all the verticesof S n − .Now we show that there are two disjoint elements of A such that bothcontain at least one vertex of S n − . Let A and A be an intersecting pair, A i ∩ V (cid:54) = ∅ , i = 1 ,
2. Consider an ( n − H that separates them in H . Then A ∩ A = H ∩ S n − since H ∩ S n − is covered by A , and it can not intersect any element of A distinct from A and A (otherwise, there would be a common point of A ∩ A and some A i ,for i (cid:54) = 1 , A ∩ A is ( n − p ∈ A , q ∈ A are arbitrary points, then let x = [ p, q ] ∩ H . We obtain[ p, x ] ⊆ A , [ x, q ] ⊆ A , so A ∪ A is convex, and therefore it can not containall vertices of S n − . So there is an A j , j (cid:54) = 1 ,
2, say A , such that A ∩ V (cid:54) = ∅ ,and either A or A , say A , is disjoint from A , because A is disjoint from H . So we found two disjoint elements of A , A and A , even if A and A were intersecting.Let us choose two disjoint elements of A , say A and A , such that A i ∩ V (cid:54) = ∅ , i = 1 ,
2. Since S n − has n ≥ A , contains at most n − S n − . Now, we pick a vertex w in A ∩ V and an edge E of S n − connecting a vertex from V ∩ A and avertex from V (cid:114) ( A ∪ { w } ) (cid:54) = ∅ . Furthermore, we may pick an A i , say A ,such that E ∩ A ∩ A (cid:54) = ∅ , since E intersects A but also has points outside A . Finally, the desired k points: Take a point u ∈ E ∩ A ∩ A , take w ∈ A ,and take further k − A i . As u is containedin E , it is contained in each facet of S n − but the two that do not contain E .These two facets contain w , and thus, every element of F contains at leastone of the points. Since, clearly, the same holds for the elements of A as well,these points indeed satify the required property. (cid:3) We ask the following question.
Problem 6.7.
For any given n ≥ and k ≥ , find a geometric charac-terization of those n -polytopes C ⊂ R n for which every C -ball convex setis k -generated. In particular, find a geometric characterization of those n -polytopes C for which every C -ball convex set is a C -spindle.
7. Stability of the operation B + C and covering intersections ofballs We consider the Levy-Markus-Gohberg-Boltyanski-Hadwiger Covering Prob-lem (also known as the Illumination Problem) for two families of convex2 Zsolt L´angi, M´arton Nasz´odi and Istv´an Talatabodies, denoted by D ( C ) and ˜ D ( C ) (see Definition 7.1, and Remark 7.2),associated to any convex body C .The covering number (see Definition 7.3) of sets of Euclidean constantwidth (members of D ( B n )) has been studied extensively. One reason for itspopularity is its connection to Borsuk’s problem on partitioning convex setsinto sets of smaller diameter. Weissbach [42] and Lassak [24] proved thatthe covering number of a set of Euclidean constant width in R is at mostsix. In [5], this result is extended to any set K obtained as the intersection ofEuclidean unit balls with the property that the set of the centers is of diameterat most one (members of ˜ D ( B n )). Recently, further bounds on the coveringnumber of sets in ˜ D ( B n ) in dimensions n = 4 , R n is at most 5 n / (4 + log n ) (cid:0) (cid:1) n/ . This result has been extended to membersof ˜ D ( B n ) as well, cf. Bezdek [3]. For surveys on covering (illumination) seeBezdek [2], Martini and Soltan [28] and Boltyanski, Martini and Soltan [9].In this section, we study the stability of bounds on the covering numberof convex sets in D ( C ) and in ˜ D ( C ). First, we prove that the operation B + C isstable in a certain sense (Proposition 7.4), and then we deduce our stabilityresults concerning covering numbers. Definition 7.1.
Let C be a convex body in R n . Let D ( C ) = { K ⊂ R n : K = B + C ( K ) } , and ˜ D ( C ) = { K ⊂ R n : K = B + C ( X ) for some X ⊂ R n with X ⊆ B + C ( X ) } . Remark 7.2.
For any convex body C , we have D ( C ) ⊆ ˜ D ( C ) . Moreover, if C = − C then D ( C ) is the family of diametrically maximal sets of diameterone in the Minkowski space (that is, finite dimensional real Banach space)with unit ball C . Since in the Euclidean space, a convex set is diametricallymaximal if, and only if, it is of constant width, it follows that D ( B n ) is the family of sets of Euclidean constant width one . Definition 7.3.
Let K ∈ R n be a convex body. The covering number (alsocalled the illumination number ) i ( K ) of K is the minimum number of positivehomothetic copies of K , with homothety ratio less than one, that cover K .For a family F of convex bodies, we set i ( F ) = max { i ( K ) : K ∈ F} . We notethat the illumination number is usually defined via the notion of illuminationby directions (or light sources), which we do not follow here — the equivalenceof those definitions with the one given here is well known, cf. [2] . The covering number is invariant under non-singular affine transforma-tions, thus it is natural to use the Banach-Mazur distance to compare twoconvex bodies K and L : d ( K, L ) = inf { λ > K ⊂ T ( L ) ⊂ λ ( K ) } , where the infimum is taken over all non-singular affine transformations T .Recall that this distance is multiplicative (the triangle inequality holds withall and Spindle Convexity with respect to a Convex Body 23multiplication instead of addition) and the distance of a convex body fromany non-singular affine image of itself is one.We phrase, informally, the problem of the stability of the covering num-ber in the following two ways. Question 1.
Fix a convex body C . If K is ‘close’ to a set L ∈ D ( C )(resp., to a set L ∈ ˜ D ( C )), does it follow that i ( K ) ≤ i ( D ( C )) (resp., i ( K ) ≤ i ( ˜ D ( C )))? Question 2.
Fix a convex body C . If D is ‘close’ to C , does it followthat i ( D ( D )) ≤ i ( D ( C )) and i ( ˜ D ( D )) ≤ i ( ˜ D ( C ))?Recall that i ( K ) = n + 1 for any smooth convex body, while i ([0 , n ) =2 n for the cube. It illustrates that the covering number, in general, may varysignificantly along arbitrarily small perturbations. However, Theorems 11 and12 provide positive answers to both questions. Theorem 11.
For every n ∈ Z + and for every convex body C ⊂ R n , there isa δ > such that if d ( K, L ) < δ for a convex body K in R n and L ∈ D ( C ) (resp., L ∈ ˜ D ( C ) ), then i ( K ) ≤ i ( D ( C )) (resp., i ( K ) ≤ i ( ˜ D ( C )) ). Theorem 12.
For every n ∈ Z + and for every convex body C ⊂ R n , there isa δ > such that if d ( C, D ) < δ for a convex body D ⊂ R n , then i ( D ( D )) ≤ i ( D ( C )) and i ( ˜ D ( D )) ≤ i ( ˜ D ( C )) . The main tool in proving these results is the following observation thatshows that the operation B + C (and, similarly, B − C ) is stable in a certain sense. Proposition 7.4.
Let C , C , . . . be a sequence of convex bodies in R n converg-ing to a convex body C in the metric space of closed convex subsets of R n equipped with the Hausdorff metric. Let X , X , . . . be a sequence of closedsets in R n converging to a set X such that the sequence B + C i ( X i ) also con-verges (to some set K ). Assume that int K (cid:54) = ∅ or int( B + C ( X )) (cid:54) = ∅ . Then K = B + C ( X ) .Proof. First, we show that int( B + C ( X )) ⊆ K . Let u ∈ R n \ K . Then, forinfinitely many k ∈ Z , there is a q k ∈ X k such that u / ∈ q k + C . By taking asubsequence, we may assume that the q k s converge to a point, say q . Clearly, q ∈ X . Moreover, we have u / ∈ q + int C . Thus, u / ∈ int( B + C ( X )).Next, we show that int K ⊆ B + C ( X ). Let u ∈ int K . Then, there is a δ > n ∈ Z , u + δB n ⊂ B + C n ( X n ). It followsthat, for all sufficiently large n ∈ Z , u ∈ B + C n ( X ). Hence, u ∈ B + C ( X ). (cid:3) We leave it as an exercise to show that the condition int( . . . ) (cid:54) = ∅ cannotbe removed.Let K n denote the space of affine equivalence classes of convex bodiesin R n endowed with the Banach-Mazur distance. We show that i : K n → Z + is upper semi-continuous. Proposition 7.5.
Let K ∈ K n . Then there is a δ > such that i ( L ) ≤ i ( K ) for any L ∈ K n with d ( K, L ) < δ . Proof.
Let m = i ( K ). Then K is covered by v + λK, . . . , v m + λK for some0 < λ < v , . . . , v m . Let δ = 1 / √ λ . We mayassume that L ⊆ K ⊆ δL . Then L is covered by v + √ λL, . . . , v m + √ λL . (cid:3) Remark 7.6.
Let C ⊂ R n be a convex body and X ⊂ R n a set for which X ⊆ B + C ( X ) . Then int (cid:0) B + C ( X ) (cid:1) (cid:54) = ∅ . To see this, one may assume that X is convex, and then show that any point in the relative interior of X is aninterior point of B + C ( X ) . It follows that the members of D ( C ) and ˜ D ( C ) arebodies.Proof of Theorem 11. By the semi-continuity of i on K n (Proposition 7.5)and the compactness of K n , it is sufficient to show that ˜ D ( C ) and D ( C )are closed subsets of K n . Let K , K , . . . be a convergent sequence of convexbodies in ˜ D ( C ). Then, by John’s theorem, each one has an affine image K (cid:48) i such that B n ⊆ K (cid:48) i ⊆ nB n . Now, K (cid:48) , K (cid:48) , . . . is a sequence of convexbodies in ˜ D ( C ) which is convergent with respect to the Hausdorff distance.By Proposition 7.4, the limit is also in ˜ D ( C ). The statement for D ( C ) easilyfollows. (cid:3) Proof of Theorem 12.
By Theorem 11, we need to prove that for any ε > δ > d ( C, D ) < δ , then for every L ∈ D ( D ) there is a K ∈ D ( C ) with d ( K, L ) < ε .Let D k be a sequence of convex bodies in R n such that d ( C, D k ) < k and let X k be a sequence of sets such that L k := B + C k ( X k ) ∈ D ( C k ).Suppose, for contradiction, that for each L k the closest member of D ( C ) isof distance at least µ >
1. By compactness, we may choose a convergentsubsequence of the X k s. By taking a subsequence again, we may assume thatthe L k s converge, too. Now, by Proposition 7.4 (using John’s theorem, asin the proof of Theorem 11), the limit of these L k s is a member of D ( C ), acontradiction. (cid:3) Acknowledgements.
We are grateful to Antal Jo´os and Steven Taschuk forthe valuable conversations that we had with them on various topics coveredin these notes.
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