Isometric and affine copies of a set in volumetric Helly results
IISOMETRIC AND AFFINE COPIES OF A SET INVOLUMETRIC HELLY RESULTS
JOHN A. MESSINA AND PABLO SOBER ´ON
Abstract.
We show that for any compact convex set K in R d and any finitefamily F of convex sets in R d , if the intersection of every sufficiently small sub-family of F contains an isometric copy of K of volume 1, then the intersectionof the whole family contains an isometric copy of K scaled by a factor of (1 − ε ),where ε is positive and fixed in advance. Unless K is very similar to a disk,the shrinking factor is unavoidable. We prove similar results for affine copiesof K . We show how our results imply the existence of randomized algorithmsthat approximate the largest copy of K that fits inside a given polytope P whose expected runtime is linear on the number of facets of P . Introduction
Helly’s theorem is a central result in combinatorial geometry [Rad21, Hel23]. Itsays that for any finite family F of convex sets in R d , if every d +1 or fewer sets from F have a nonempty intersection, then F has a nonempty intersection. This theoremhas many extensions and applications in discrete geometry, topological combina-torics, and computational geometry (see, e.g., [ADLS17, HW17, DLGMM19] andthe references therein).In the quantitative versions of Helly’s theorem, we aim to characterize finitefamilies of convex sets whose intersection is quantifiably large rather than simplynonempty. B´ar´any, Katchalski, and Pach started this direction of research whenthey proved a volumetric version of Helly’s theorem [BKP82,BKP84]. They showedthat if the intersection of any d or fewer elements of a finite family of convex setsin R d has volume greater than or equal to one, the intersection of the whole familymust be nonempty and have volume greater than or equal to d − d . The guarantee on the volume of the intersection is smaller than the bound weask in the 2 d -tuples. Even though this volume loss has been reduced significantly[Nas16, Bra17], it is unavoidable even if we are willing to check much larger sub-families [DLLHRS17]. A way to obtain exact quantitative Helly-type theorems, inwhich no such loss is present, is to impose additional conditions.Given a family W of sets in R d , we can ask if there exists a positive integer n such that for any finite family F of convex sets in R d , if the intersection of any n or fewer of them contains a set of W of volume one, then (cid:84) F contains a set of W of volume one. Sarkar, Xue, and Sober´on recently showed that such a result holdsfor various families W , including the family of all axis-parallel boxes or the familyof all ellipsoids [SXS19]. We say that W acts as a family of witness sets. If such astatement holds, we say that W admits an exact Helly theorem for the volume.In this manuscript, we explore whether certain new families W admit an exactHelly theorem for the volume. Except for the result for ellipsoids, the familiesconsidered in [SXS19] fix the orientation of the sets of W . We are interested infamilies of witness sets where the orientation is not fixed. Given a convex set K in This research project was done as part of the 2020 Baruch Discrete Mathematics REU, sup-ported by NSF awards DMS-1802059, DMS-1851420, and DMS-1953141. a r X i v : . [ m a t h . M G ] O c t MESSINA AND SOBER ´ON
Figure 1.
Consider the three great circles in S formed by the xy -plane, the yz -plane and the xz -plane. Let K be the convexhull of their union. Theorems 1.0.1 and 1.0.2 show that neither W iso ( K ) or W aff ( K ) admit an exact Helly theorem for the volume. R d , we consider the families W aff ( K ) = { a + αAK : A ∈ SL ( d ) , α ∈ R , a ∈ R d } and W iso ( K ) = { a + αAK : A ∈ O ( d ) , α ∈ R , a ∈ R d } of affine or scaled isometric copies of K . For scaled isometric copies, we show thatunless K has a sufficiently large intersection with the boundary of the minimumvolume ball containing it, W iso ( K ) does not admit an exact Helly theorem forthe volume. For affine copies, a similar statement holds for the minimum volumeenclosing ellipsoid of K . For a compact set K ⊂ R d , let B ( K ) be the minimumvolume ball such that K ⊂ B ( K ). We denote by ∂B ( K ) the boundary of B ( K ). Theorem 1.0.1.
Let K be a compact set in R d . If there is a closed half-sphere D ⊂ ∂B ( K ) such that K ∩ D has measure under the Haar measure of ∂B ( K ) ,then W iso ( K ) does not admit an exact Helly theorem for the volume. Theorem 1.0.2.
Let K be a compact set in R d such that B ( K ) is also the minimumvolume ellipsoid containing K . If there is a closed half-sphere D ⊂ ∂B ( K ) suchthat K ∩ D has measure under the Haar measure of ∂B ( K ) , then W aff ( K ) doesnot admit an exact Helly theorem for the volume. For any bounded set K in R d , we may assume B ( K ) is its minimum enclosingellipsoid by applying a particular affine transformation, so no generality is lost withthe condition of Theorem 1.0.2. Theorem 1.0.1 implies that a convex set K must bevery similar to a ball for W iso ( K ) to admit an exact Helly theorem for the volume.Theorem 1.0.2 implies that a convex K must be very similar to an ellipsoid for W aff ( K ) to admit an exact Helly theorem for the volume.In many recent results regarding volumetric Helly-type theorems, analytic prop-erties of ellipsoids are key ingredients of the proofs [Nas16, Bra16, Bra17, Bra18,FVGM20, DFN19]. Results on the sparsification of John decompositions of theidentity can be translated to Helly-type theorems. Theorems 1.0.1 and 1.0.2 showthat the study of ellipsoids is much more intertwined with volumetric Helly-typetheorems than previously thought. The techniques we use to prove theorems 1.0.1and 1.0.2 involve the probabilistic method. These methods can also be used to SOMETRIC AND AFFINE COPIES OF A SET IN VOLUMETRIC HELLY RESULTS 3 give a lower bound for Helly numbers for sets where those theorems fail to apply.Theorems 1.0.1 and 1.0.2 apply to any polytope K . There are more general sets towhich they apply, as Figure 1 shows.On a positive note, we show that if we accept a loss of ε on the volume, we dohave such Helly-type theorems. Theorem 1.0.3.
Let d be a positive integer and ε > . Let K be a compact set witha nonempty interior. There exists an integer n = n ( K, ε ) such that the followingstatement holds. If F is a finite family of convex sets such that the intersectionof any n or fewer sets of F contains a set of W iso ( K ) of volume one, then ∩F contains a set of W iso ( K ) of volume − ε . The theorem above has consequences in computational geometry. The prob-lem of finding the largest copy of a polygon inside another is interesting, andmany algorithms have been constructed to solve instances of this problem [Ame94,AAS98, CCKS16, HHKK + ε -approximation of the largest scaled isometric copy of a polytope K inside anotherpolytope P is a linear-programming type (LP-type) problem. Therefore, it can besolved by randomized algorithms in expected linear time in terms of the numberof facets of P (with hidden factors depending on K and ε , which we assume arefixed).As an example, consider the problem of finding the largest volume hypercubecontained inside a polytope P ⊂ R d (we consider a hypercube as an isometricscaled copy of [0 , d ). If we insist that the hypercube is axis-parallel, the resultsof Sarkar et a. show that this is an LP-type problem [SXS19]. If we allow it tohave any orientation, the results here show that approximating the largest hyper-cube is an LP-type problem, but there is no associated Helly theorem for an exactcomputation.We set preliminaries and notation in Section 2 and prove Theorems 1.0.1 and1.0.2 in Section 3. We prove Theorem 1.0.3 in Section 4, where we also discussthe computational applications. Finally, some future directions of research arepresented in Section 5. 2. Preliminaries and notation
Let B d be the unit ball in R d , and let S d − be its boundary. Further, let O ( d )denote the group of orthogonal d × d matrices, and SL ( d ) denote the group of d × d matrices of determinant 1 . We denote by µ the Haar probability measure on S d − ,which is invariant under O ( d ). We denote by (cid:104)· , ·(cid:105) the standard dot product in R d .For a unit vector u , we say that D ( u ) ⊂ S d − is the half-sphere with direction u if D ( u ) = { x ∈ S d − : (cid:104) x, u (cid:105) ≥ } . For ε >
0, we define the ε -neighborhood of D ( u ) as the set D ( u ) ε = { x ∈ S d − : (cid:104) x, u (cid:105) > − ε } .For a unit vector u , we denote by H u the closed half-space that contains B d andwhose boundary contains u . In other words H u = { x ∈ R d : (cid:104) x, u (cid:105) ≤ } . For a compact set M ⊂ B d and λ > F ( M, λ ) = { H u : dist( u, M ) ≥ λ } . The distance above is computed using the Euclidean distance in R d . Given a d × d matrix A and a set K ⊂ R d , we denote by AK the set { Ax : x ∈ K } . Wecan parametrize W iso ( K ) by triples ( a, α, A ) ∈ R d × R × O ( d ). The triple ( a, α, A )corresponds to the set a + αAK . A set may be represented many times if K = AK for more than one matrix A in O ( d ) or SL ( d ). For a compact set M ⊂ R d we define S ( M ) = { ( a, α, A ) ∈ R d × R × O ( d ) : a + αAK ⊂ M } . MESSINA AND SOBER ´ON
By checking subsequences it is simple to note that if M is compact, S ( M ) iscompact. Definition 1.
Let W be a family of sets in R d . We denote by h ( W ) the smallestpositive integer n , if it exists, such that the following holds. For any finite familyof convex sets in R d , if the intersection of n or fewer of them contains a set of W ,then the intersection of the whole family contains a set of W . If no such n exists,we say h ( W ) = ∞ . We say that h ( W ) is the Helly number for W . The family W admits an exactHelly theorem for the volume if h ( W (cid:48) ) < ∞ for W (cid:48) = { W ∈ W : vol( W ) ≥ } .3. Isometric and affine copies
In order to prove Theorem 1.0.1, it suffices to construct for each positive integer n a finite family of convex sets whose intersection does not contain an element of W iso ( K ) of volume 1, but the intersection of any n sets does. Lemma 3.0.1.
Let n, d be positive integers. Let K ⊂ B d be a compact set suchthat µ ( K ∩ S d − ) < /n . There exists a positive constant δ = δ ( n, d, K ) > suchthat the following holds. For any set L of at most n points in S d − , there exists A ∈ O ( d ) such that dist( L, AK ) ≥ δ. Proof.
Let ( x , . . . , x n ) ∈ ( S d − ) n be an n -tuple of points in S d − . If we pick arandom matrix A ∈ O ( d ), then the probability P [ x ∈ AK ] = µ ( K ∩ S d − ) < n . By a simple union bound, there exists a matrix A ∈ O ( d ) such that none of x , . . . , x n are contained in AK . Let f : (cid:0) S d − (cid:1) n → R ( x , . . . , x n ) (cid:55)→ max A ∈ O ( d ) dist ( { x , . . . , x n } , AK ) . We know that f ( x , . . . , x n ) > x , . . . , x n ) ∈ (cid:0) S d − (cid:1) n . The function f is also continuous. Since the domain is compact, the function attains a minimumvalue δ . This is the constant we were looking for. (cid:3) Lemma 3.0.2.
Let K ⊂ B d and n be a positive integer. Suppose there exists ahalf-sphere D ⊂ S d − such that µ ( K ∩ D ) < /n . Then, there exists a positiveconstant δ = δ ( n, d, K ) > such that for any collection H , . . . , H n of n half-spaces, each containing the unit ball, their intersection contains an isometric copyof K scaled by a factor of δ . An intuitive illustration of the proof below is presented in Figure 2.
Proof.
Let u be the unit vector such that D = D ( u ). We can find a positive ε such that µ ( K ∩ D ε ( u )) < /n . Let x i be the contact point of B d and H i .Consider the set K ε = { x ∈ K : (cid:104) x, u (cid:105) ≥ − ε } . By Lemma 3.0.1, we know there exists a δ > T such thatdist( { x , . . . , x n } , T K ε )) ≥ δ . Therefore, for each i we have H i ∈ F ( T K ε , δ ) , SOMETRIC AND AFFINE COPIES OF A SET IN VOLUMETRIC HELLY RESULTS 5 x x x x x x uK x x x x x x AK Au x x x x x x AuA ( K + λu ) Figure 2.
We show the idea behind the proof of Lemma 3.0.2.The part in red is the complement to D ( u ). We rotate K so that K ∩ D is far from any x i , and then translate it in the direction of u . At that point, it’s possible to scale K up while remaining insidethe intersection of the half-spaces tangent at x i on the sphere.where F ( T K ε , δ ) is the family defined in Section 2. If we consider Q = (cid:84) F ( K ε , δ ),we have n (cid:92) i =1 H i ⊃ T Q.
Therefore, it suffices to show that Q contains an isometric copy of K scaled by afactor greater than one. Let λ = (1 /
2) min { ε , δ } . The translate K + λu is in theinterior of Q . Therefore, there exists a constant δ > δ )( K + λu ) ⊂ Q, as required. (cid:3) Proof of Theorem 1.0.1.
We may assume without loss of generality that B ( K ) = B d . For each u ∈ S d − , let M u be a simplex that contains B d and is tangent to B d at u . Let F = { M u : u ∈ S d − } . For a fixed positive integer n , let us use F to show that W iso ( K ) does not admitan exact Helly theorem for the volume. Take δ = δ ( n ( d + 1) , d, K ) from Lemma3.0.2. For each M u ∈ F , let S ( M u ) = { ( a, α, A ) ∈ R d × R × O ( d ) : a + αAK ⊂ M u , | α | = 1 + δ } . Since M u is compact, S ( M u ) is compact. The intersection of any n sets of F isa polytope of at most n ( d + 1) facets that contains B d , and therefore contains anisometric copy of (1+ δ ) K . In other words, every n sets of the family G = { S ( M u ) : M u ∈ F} have a nonempty intersection. The family G has an empty intersectionsince (cid:84) F = B d and, by construction, B d does not contain a scaled isometric copyof K with a factor greater than 1. Since the elements of G are compact, there mustbe a finite family G (cid:48) whose intersection is empty. Let F (cid:48) ⊂ F be the family thatcorresponds to G (cid:48) . We know that • F (cid:48) is finite, • (cid:84) F (cid:48) does not contain an isometric copy of K scaled by a factor of 1 + δ ,and • the intersection of every n or fewer sets of F (cid:48) contains an isometric copy of K scaled by a factor of 1 + δ .Since we can construct such a family of each positive integer n , the family W iso ( K )does not admit an exact Helly theorem for volume. (cid:3) MESSINA AND SOBER ´ON Kα Figure 3.
An illustration of Example 3.0.3. If α > π − πn , then W iso ( K ) does not admit an exact Helly theorem for the volumewith Helly number smaller than n + 1. Proof of Theorem 1.0.2.
We may assume without loss of generality that B d is theminimum volume ellipsoid containing K . Let F be the same family of convex setsas in the proof of Theorem 1.0.1. We also denote its elements by M u . For any affinecopy K (cid:48) of K contained in B d , let us look at the minimum volume ellipsoid E ( K (cid:48) )containing K (cid:48) . If E ( K (cid:48) ) (cid:54) = B d , since K (cid:48) ⊂ B d , we know E ( K (cid:48) ) has a smaller volumethan B d . Therefore, the affine function that sends E ( K (cid:48) ) to B d must increasevolume.In other words, the largest volume that an affine copy of K contained in B d can have is vol( K ). The affine transformation associated with an affine copy of K of maximal volume in B d must preserve B d , and therefore be an isometry. Let n be a fixed positive integer. By the arguments of the proof of Theorem 1.0.1, theintersection of any n or fewer sets of F contains a scaled isometric copy of K by afactor of 1 + δ .Consider the sets of the form S ( M u ) = { ( a, α, A ) ∈ R d × R × SL ( d ) : a + αAK ⊂ M u , | α | = 1 + δ } . Even though SL ( d ) is not compact, every set S ( M u ) is compact. This followsfrom the fact that M u is compact, so the set of matrices A in the third coordinateof a point ( a, α, A ) in S ( M u ) is bounded. The same argument as before allows usto extract a finite subfamily F (cid:48) ⊂ F whose intersection does not contain an affinecopy of K of volume (1 + δ ) d vol( K ). However, the intersection of any n sets of F (cid:48) does contain such an affine copy of K . Since this can be done for any n , then W aff ( K ) does not admit an exact Helly theorem for the volume. (cid:3) Example 3.0.3.
Let C ⊂ S d − be a circular cap of S d − of measure greater than − n . Consider K = conv( S d − \ C ) . The arguments in the proof of Theorem1.0.1 show that, if W iso ( K ) admits an exact Helly theorem for the volume, the Hellynumber must be at least n + 1 . See Figure 3 for an illustration in dimension two. Problem 3.0.4.
For K as in Example 3.0.3, does W iso ( K ) admit an exact Hellytheorem for the volume? Approximations and computational applications
The results of this section are a consequence of the following simple lemma.
SOMETRIC AND AFFINE COPIES OF A SET IN VOLUMETRIC HELLY RESULTS 7
Lemma 4.0.1.
Let W and W be families of sets in R d , each of which has a finiteHelly number. Then, W ∪ W also has a finite Helly number and h ( W ∪ W ) ≤ h ( W ) + h ( W ) . Proof.
We prove the contrapositive. Let F be a finite family of convex sets such thattheir intersection does not contain a set of W ∪ W . Then, since the intersectiondoes not contain a set of W , we can find a subfamily F ⊂ F of cardinality at most h ( W ) whose intersection does not contain a set of W . Analogously, we can find asubfamily F ⊂ F of cardinality at most h ( W ) whose intersection does not containan element of W . The family F ∪ F has at most h ( W ) + h ( W ) elements, andits intersection contains no element of W ∪ W . (cid:3) Lemma 4.0.2.
Let K be a compact convex set in R d with a nonempty interior,and ε > be a constant. Then, we can find a positive integer t and t matrices A , . . . , A t in O ( d ) such that every isometric copy of K contains a translate of oneof the sets (1 − ε ) A i K .Proof. Let K c be the closure of the complement of K . We assume without loss ofgenerality that the origin is in the interior of K . Then, (1 − ε ) K is contained inthe interior of K , so dist((1 − ε ) K, K c ) > . Consider the function f : O ( d ) → R A (cid:55)→ max a ∈ R d dist( a + A ((1 − ε ) K ) , K c ) . This function is continuous. The set M = f − ((0 , ∞ ]) ⊂ O ( d ) is open andcontains the identity. For each A ∈ O ( d ), consider the set M A = { BA − : B ∈ M } . The family M = { M A : A ∈ O ( d ) } is an open cover of O ( d ). Since O ( d ) iscompact, there exists a finite collection A , . . . , A t of matrices in O ( d ) such that M A , . . . , M A t cover O ( d ). Let DK be an isometric copy of K , for some D ∈ O ( d ).Since D − ∈ O ( d ), there is an A i such that D − ∈ M A i .In other words, D − A i ∈ M . Therefore, there exists an a ∈ R d such thatdist( a + D − A i ((1 − ε ) K ) , K c ) > , which is equivalent to dist( Da + A i ((1 − ε ) K ) , DK c ) > . Finally, this means that there is a translate of A i ((1 − ε ) K ) contained in the interiorof DK . (cid:3) Given two compact convex sets K and P in R d , an interesting problem is tofind the largest scaled isometric copy of K contained in P . If K is fixed and P is a polytope with n facets, we would like to know the complexity of solving thisproblem in terms of n . Formally, we want to compute the constant α ( K, P ) = max { α : a + αAK ⊂ P for some a ∈ R d , A ∈ O ( d ) } We show how to use Lemma 4.0.2 to find an approximation of this parameter.First, if we are given a particular set A , . . . , A t of matrices in O ( d ) we can definea similar parameter β ( K, P ) = max { α : a + αA i K ⊂ P for some a ∈ R d , ≤ i ≤ t } . MESSINA AND SOBER ´ON
Theorem 4.0.3.
Let K be a compact convex set in R d with a nonempty interior.Let A , . . . , A t be matrices in O ( d ) used to define the parameter β ( K, · ) . For a finitefamily F of convex sets in R d , there exists a subfamily F (cid:48) of cardinality t ( d + 1) such that β (cid:16) K, (cid:92) F (cid:17) = β (cid:16) K, (cid:92) F (cid:48) (cid:17) . Proof.
We use an argument similar to the one in the proof of Lemma 4.0.1. Foreach i = 1 , . . . , t , define the parameter β i ( K, · ) as β i ( K, P ) = max { α : a + αA i K ⊂ P for some a ∈ R d } . Let β i = β i ( K, (cid:84) F ). By Helly’s theorem for translates of a set, we know that thefamily of witness sets W i = { a + β i A i K : a ∈ R d } has Helly number at most d + 1. Therefore, there is a family F i ⊂ F of size d + 1such that β i ( K, (cid:84) F i ) = β i . Since β ( K, · ) = max ≤ i ≤ t β i ( K, · ), it suffices to take F (cid:48) = (cid:83) ti =1 F i to finish the proof. (cid:3) The theorem above shows that computing β ( K, P ) is an LP-type problem. Con-sider d, t to be fixed. To compute β ( K, P ) for a polytope P , we first write P asthe intersection of n half-spaces, P = ∩ ni =1 H i . A brute-force algorithm would be asfollows. For a fixed t ( d + 1)-tuple I ⊂ [ n ], let P (cid:48) = ∩ i ∈ I H i . We compute β ( K, P (cid:48) ).We repeat this for all (cid:0) nt ( d +1) (cid:1) = O ( n t ( d +1) ) different t ( d + 1)-tuples of half-spaces,and output the minimum number found.We can do better by applying a randomized algorithm, such as the randomizeddual-simplex algorithm [SW92] that runs in O ( n ) expected time. The parameters t, d affects the hidden constant factor, but not the dependence on n . The algorithmsdepend on access to an oracle that finds β ( K, P (cid:48) ) when P (cid:48) is the intersection of t ( d + 1) half-spaces. We discuss below why such a computation is possible when K is a polytope. First, let us show how the computation of β ( K, P ) implies anapproximation of α ( K, P ). Corollary 4.0.4.
Let K be a convex polytope in R d whose interior is not empty,and ε > be fixed. There is a randomized algorithm that runs in O ( n ) expectedtime such that approximates α ( K, P ) up to a relative error of ε .Proof. Let A , . . . , A t be the matrices from Lemma 4.0.2. Then, for any polytope P we have (1 − ε ) α ( K, P ) ≤ β ( K, P ) ≤ α ( K, P ) . In other words, β ( K, · ) approximates α ( K, · ) with a relative error not greater than ε . We can run the randomized dual-simplex algorithm and find β ( K, P ) in expected O ( n ) time. (cid:3) If K and P are polytopes, we can check if a + β i K ⊂ P by checking the verticesof K one by one. A maximal translate of K in P will have contact points withfacets of P whose normal vectors capture the origin.Let us look at the example of approximating the size of the largest equilateraltriangle inside a polytope. The first task, finding the value of t , can be done byfinding the angle α at which any rotation of an equilateral triangle of side 1 − ε fitsinside an equilateral triangle of side 1 (see Figure 4). We set t = π/α . The rotations A , . . . , A t are simply rotations by an angle of πjt for j = 1 , . . . , t . Once A , . . . , A t are fixed, we follow the algorithms described above. In the plane, a similar processcan be done for any convex polygon K .In high dimensions, the problem of computing t and the matrices A , . . . , A t isinteresting. If we consider O ( d ) as a metric space, a sufficiently dense net depending SOMETRIC AND AFFINE COPIES OF A SET IN VOLUMETRIC HELLY RESULTS 9 α − ε τ − τ Figure 4.
To find the largest angle α for which a rotated copy ofan equilateral triangle of side 1 − ε fits inside a side 1 equilateraltriangle, it suffices to use the law of sines twice in the blue triangleand solve for α .on ε and K will work. The precise value of t would not affect the expected time interms of n for the algorithms mentioned above. However, those algorithms carryhidden factors in terms of the combinatorial complexity of the LP-type problem,which is t ( d + 1). The following problem is relevant. Problem 4.0.5.
Given a polytope K in R d and ε > , compute the smallest value t such that there exist A , . . . , A t ∈ O ( d ) for which any isometric copy of K containsa translate of A i ((1 − ε ) K ) for some ≤ i ≤ t . Future Directions of Research
In this work, we address the problem of finding for which sets K the collections W iso and W aff admit an exact Helly theorem for the volume. Theorem 1.0.1 showsthat if, for a given convex set K, W iso admits an exact Helly theorem for thevolume, then K must have a large intersection with its minimal enclosing sphere.If a negative answer to Problem 3.0.4 holds, one may ask the following questions. Problem 5.0.1. Is B d the only set for which W iso admits an exact Helly theoremfor the volume? Problem 5.0.2.
Are ellipsoids the only sets for which W aff admits an exact Hellytheorem for the volume? One may alternatively ask which collections of copies of a given set K admit anexact Helly theorem for the volume. In particular, for a set K ⊆ R d and a subgroup G < O ( d ), one may ask whether the set W G = { a + αAK | A ∈ G, α ∈ R , a ∈ R d } admits an exact Helly theorem for the volume. Theorem 1.0.1 shows that if G = O ( d ) , a negative answer holds unless K is very similar to a ball. Lemma 4.0.1implies that for any finite subgroup G and any compact set K of positive volume,the set W G admits an exact Helly theorem for the volume. For infinite subgroups,the following problem remains open. Problem 5.0.3.
Given a subgroup
G < O ( d ) , for which convex sets K ⊂ R d does W G admit an exact Helly theorem for the volume? In particular, does K have to be similar to a G -invariant subset of R d as inTheorem 1.0.1? 6. Acknowledgments
The authors thank Edgardo Rold´an-Pensado for making Figure 1.
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