AA M´ELANGE OF DIAMETER HELLY-TYPE THEOREMS
TRAVIS DILLON AND PABLO SOBER ´ON
Abstract.
A Helly-type theorem for diameter provides a bound on the di-ameter of the intersection of a finite family of convex sets in R d given someinformation on the diameter of the intersection of all sufficiently small subfam-ilies. We prove fractional and colorful versions of a longstanding conjectureby B´ar´any, Katchalski, and Pach. We also show that a Minkowski norm ad-mits an exact Helly-type theorem for diameter if and only if its unit ball isa polytope and prove a colorful version for those that do. Finally, we proveHelly-type theorems for the property of “containing k colinear integer points.” Introduction
Helly’s theorem is one of the most prominent results on the intersection prop-erties of families of convex sets [21, 29]. It says that if the intersection of every d + 1 or fewer elements of a finite family of convex sets in R d is nonempty, thenthe intersection of the entire family is nonempty. This result has many extensionsand generalizations, including topological, colorful, and fractional variants (see, forexample, [2, 22] and the references therein).Quantitative versions of Helly’s theorem guarantee that the intersection of afamily of convex sets is not just nonempty but “large” in some quantifiable sense.B´ar´any, Katchalski, and Pach initiated this direction of research [7, 8] when theyproved that if the intersection of every d or fewer elements of a finite family ofconvex sets in R d has volume greater than or equal to 1, then the intersection ofthe entire family has volume at least d − d . Nasz´odi [28] improved the guarantee of the volume in the intersection to d − d ,mostly settling this volumetric variant. His approach, based on sparsification ofJohn decompositions of the identity, has been improved in several articles [11, 12,14,20]. A constellation of related results that adjust the function measuring the sizeof the intersection, the cardinality of the subfamily intersection, or the guaranteein the conclusion have since been proven [16, 30, 31, 33]. B´ar´any, Katchalski, andPach conjectured a Helly-type theorem for diameter as well, which remains open. Conjecture 1.1 (B´ar´any, Katchalski, Pach 1982 [7]) . Let F be a finite familyof convex sets in R d . If the intersection of every d or fewer members of F hasdiameter greater than or equal to 1, then the intersection of F has diameter greaterthan or equal to cd − / , for some absolute constant c > . B´ar´any, Katchaslki, and Pach showed that the diameter of the intersection isat least d − d . Brazitikos improved this to d − / , which is the first polynomialbound on the diameter of the intersection [13]. Brazitikos [12] also proved, underthe hypothesis that the intersections of subfamilies of size αd have diameter at least1 (for some large enough absolute constant α ), that the intersection of the entirefamily has diameter at least cd − / and strengthened the bound to cd − / under the This research project was completed as part of the 2020 Baruch Discrete Mathematics REU,supported by NSF awards DMS-1802059, DMS-1851420, and DMS-1953141. Sober´on’s researchis also supported by PSC-CUNY grant 62639-00-50. a r X i v : . [ m a t h . M G ] S e p DILLON AND SOBER ´ON further assumption that each set is centrally symmetric. Asymptotically optimalbounds are known for much larger subfamily intersection sizes [33].In this manuscript we prove several new Helly-type theorems for the diameter.First, we prove that Conjecture 1.1 holds for at least a large subfamily of F . Theorem 1.2.
There exists a decreasing function γ : (0 , √ → (0 , such that γ ( c ) → as c → and the following holds for every c ∈ (0 , √ , α ∈ (0 , , and d ≥ . Let β = 1 − (cid:0) − α · γ ( c ) (cid:1) / d and F be a finite family of convex sets in R d . If (cid:84) H has diameter greater than or equal to 1 for at least α (cid:0) |F| d (cid:1) subcollections H ⊆ F of d sets, then there exists a subfamily G ⊆ F such that |G| ≥ β |F| andthe diameter of (cid:84) G is greater than or equal to cd − / . We exclude the case d = 1 since the real line has an exact diameter Helly-typetheorem, while all higher dimensions do not. Crucially, for α = 1 we have β → c →
0, so the size of the subfamily can be arbitrarily close to that of the original set.Theorem 1.2 suggests that it should be possible to extend the diameter conjectureto a fractional version (with an absolute c ) in the same way that Katchaslki andLiu generalized Helly’s theorem [26].We also prove a colorful variant of Theorem 1.2, similar to Lov´asz’s colorful Hellytheorem [5], at the cost of a slightly smaller constant β . Theorem 1.3.
Let γ : (0 , √ → (0 , be the function in Theorem 1.2. For each c ∈ (0 , √ ), α ∈ (0 , , and d ≥ , set β = 1 − d (cid:0) − α · γ ( c ) (cid:1) / d . Assumethat F , . . . , F d are finite families of convex sets in R d and set N = (cid:81) di =1 |F i | . If (cid:84) di =1 F i has diameter greater than or equal to 1 for at least α (cid:0) N d (cid:1) different d -tuples ( F i ) di =1 with F i ∈ F i for each i , then there exists an index k ∈ [2 d ] and a subfamily G ⊆ F k such that |G| ≥ β |F k | and the diameter of (cid:84) G is greater than or equal to cd − / . The description “colorful” is derived from thinking of each family F i as having aparticular color. Then, if the intersections of sufficiently many colorful collections(containing one set of each color) have large diameter, there is a large monochro-matic family whose intersection has large diameter. In light of Theorem 1.3, wepostulate a colorful version of the B´ar´any-Katchalski-Pach conjecture. Conjecture 1.4.
Let F , . . . , F d be finite families of convex sets in R d . If (cid:84) di =1 F i has diameter greater than or equal to 1 for every d -tuple ( F i ) di =1 with F i ∈ F i , thenthere exists an index k ∈ [2 d ] such that (cid:84) F k has diameter greater than or equal to cd − / , for some absolute constant c . Any Helly-type theorem for diameter necessarily entails some loss—it is notpossible to conclude that intersection of the entire family has diameter at least 1even by checking arbitrarily large subfamilies [33]. Sarkar, Xue, and Sober´on [31]suggested that this may be a consequence of the norm used to measure diameter,and that the (cid:96) norm may give exact Helly-type diameter results. We show thatthis is indeed the case. Theorem 1.5.
Let ρ be a Minkowski norm in R d whose unit ball is a polytope with k facets, and let F be a finite family of convex sets in R d . If the intersection ofevery kd or fewer members of F has ρ -diameter greater than or equal to 1, then (cid:84) F has ρ -diameter greater than or equal to 1. Moreover, this statement is not trueif kd is replaced by kd − . In particular, there is an exact diameter Helly-type theorem in the (cid:96) -norm,although the intersection condition on subfamilies is necessarily exponential. We M´ELANGE OF DIAMETER HELLY-TYPE THEOREMS 3 present three proofs of Theorem 1.5, one of which implies a colorful version (seeTheorem 3.4). In Theorem 3.7, we prove that no other Minkowski norm admits anexact Helly-type theorem for diameter, thus characterizing the norms for which anexact theorem is possible.The particular case of the (cid:96) ∞ norm implies a different relaxation of Conjecture1.1. Corollary 1.6.
Let F be a finite family of convex sets in R d . If the intersectionof every d or fewer elements of F has diameter greater than or equal to 1, then (cid:84) F has diameter greater than or equal to d − / . If we relax Conjecture 1.1 to checking subfamilies of quadratic cardinality in thedimension, then an application of N´aszodi’s method guarantees that the diameterof (cid:84) F is at least d − (see, e.g., [20, Theorem 1.4]). To obtain a bound of d − / ,however, the method would require that each set be centrally symmetric.Finally, we investigate a discrete analogue of diameter Helly-type theorems.Doignon extended Helly’s theorem to the integer lattice [19], showing that if theintersection of every d or fewer elements of a finite family of convex sets in R d contains an integer point, then the entire intersection also contains an integer point. This result was proved independently by Bell [10] and by Scarf [32]. In most cases,the aim of quantitative Helly-type theorems for the integer lattice is to bound thenumber of integer points in the intersection of a family of convex sets [1, 3, 17, 18].Such work can be thought of as Helly-type theorems for “discrete volume”. Wethink of a convex set as having large “discrete diameter” if it contains many colinearinteger points. In contrast to most continuous diameters, there is an exact Helly-type theorem for discrete diameter.
Theorem 1.7.
Let k be a positive integer and F be a finite family of convex setsin R d . If the intersection of every d or fewer elements of F contains k colinearinteger points, then (cid:84) F contains k colinear integer points. Our proof also implies a colorful version of Theorem 1.7. Doignon’s theoremshows that the size of the subfamilies in the hypothesis is necessarily exponentialin the dimension, but this size can be significantly reduced if it suffices to maintaina bound on the diameter of a large subfamily of F . Corollary 1.8.
For every positive integer d and real number α ∈ (0 , , there existsa real number β = β ( α, d ) > such that the following holds. Assume that F is afinite family of convex sets in R d and let k be a positive integer. If (cid:84) H contains atleast k colinear integer points for at least α (cid:0) |F| d +1 (cid:1) subcollections H ⊆ F of d + 1 sets, then there exists a subfamily G ⊆ F such that |G| ≥ β |F| whose intersectioncontains k colinear integer points. Since Doignon’s theorem is optimal, the proportion β (1 , d ) is necessarily strictlyless than 1.We present the proof of Theorem 1.7 in Section 2. Our results for Minkowskinorms are collected in Section 3, and Section 4 contains the proofs of theorems 1.2and 1.3. 2. Discrete diameter results
The proofs in this section employ similar methods to those of Sarkar, Xue, andSober´on in [31], in which a suitable parametrization reduces quantitative Helly-typetheorems to standard Helly-type theorems in higher-dimensional spaces.We denote the standard inner product in R d by (cid:104)· , ·(cid:105) . DILLON AND SOBER ´ON
Proof of Theorem 1.7.
Let v ∈ R d be a vector whose components are algebraicallyindependent. In particular, (cid:104) v, z (cid:105) (cid:54) = 0 for every z ∈ Z d \ { } . For every convex set K ⊆ R d , we define the set S ( K ) = (cid:8) ( x, y ) ∈ R d × R d : x ∈ K, x + ( k − y ∈ K, (cid:104) v, y (cid:105) > (cid:9) , which is convex. Moreover, if x ∈ K , x + ( k − y ∈ K , and (cid:104) y, v (cid:105) (cid:54) = 0, then either( x, y ) ∈ S ( K ) or ( x + ( k − y, − y ) ∈ S ( K ). Now consider the family G = { S ( K ) : K ∈ F} . The conditions of the theorem imply that the intersection of every 2 d or fewersets in G contains a point of Z d in their intersection. By Doignon’s theorem, (cid:84) G contains an integer point. If ( x, y ) is such a point, then y (cid:54) = 0 and the k colinearinteger points x, x + y, . . . , x + ( k − y are contained in every member of F . (cid:3) The proof above is quite malleable. For example, replacing Doignon’s theoremwith its colorful version (proved by De Loera, La Haye, Oliveros, and Rold´an-Pensado [15]) yields a colorful version of Theorem 1.7. To obtain Corollary 1.8, wereplace Doignon’s theorem by the following fractional version.
Theorem 2.1 (B´ar´any, Matouˇsek 2003 [9]) . For every positive integer d and realnumber α ∈ (0 , there exists a real number β = β ( α, d ) > such that the followingis true. If F is a finite family of convex sets such that (cid:84) H contains an integer pointfor at least α (cid:0) |F| d +1 (cid:1) subcollections H ⊆ F of d + 1 sets, then there is a subfamily G ⊆ F with at least β |F| sets whose intersection contains an integer point. It is unclear whether the number 4 d in Theorem 1.7 is optimal. Since the case k = 1 is Doignon’s theorem, 4 d cannot be replaced by anything smaller than 2 d .The following construction, which generalizes a construction communicated by Gen-nadiy Averkov for d = 2, improves the lower bound to d d . Claim 2.2.
For each d ≥ , there exists a finite family F of convex sets in R d suchthat the intersection of any d d − sets in F contains 3 colinear integer points but (cid:84) F does not.Proof. Let R ⊆ { , , , } d be the collection of integer points where exactly onecoordinate is in { , } and let Q = { , } d . That is, Q ∪ R is the set of integerpoints in the hypercube [0 , d that do not lie on its ( d − F = { conv( Q ∪ R \ { x } ) : x ∈ R } , which is a collection of d d sets. The intersection of any d d − F containsevery point in Q and one point in R , so it contains 3 colinear integer points. Butthe integer points in (cid:84) F are exactly those in Q , which does not contain 3 colinearinteger points. (cid:3) Diameter results for Minkowski norms
Definition 3.1.
Let K ⊆ R d be a compact convex set with nonempty interior thatis symmetric about the origin. The Minkowski norm of K is defined by ρ K ( x ) = min { t ≥ x ∈ tK } . Any Minkowski norm is a norm in the usual sense. Given a vector v ∈ R d \ { } ,the v -width of a compact convex set K ⊆ R d is max x,y ∈ K (cid:104) x − y, v (cid:105) . Let K ⊆ R d polytope with k facets, and for each facet L i , let v i be the vector such that { x ∈ R d : (cid:104) x, v i (cid:105) = 1 } is the hyperplane containing L i . We assume that L i and L ( k/ i are opposing facets, so v ( k/ i = − v i . If ρ is the Minkowski norm whose unit ballis K , then ρ ( x ) = max ≤ i ≤ k/ |(cid:104) x, v i (cid:105)| (3.1) M´ELANGE OF DIAMETER HELLY-TYPE THEOREMS 5 for every x ∈ R d .The proof in Section 2 can be adapted to simplify the proof of a Helly-typetheorem for v -width by the second author [33]. Theorem 3.2.
Let v be a nonzero vector in R d and F be a finite family of compactconvex sets in R d . If the intersection of every d sets in F has v -width greater thanor equal to 1, then (cid:84) F has v -width greater than or equal to 1.Proof. For every convex set K ⊆ R d , let S ( K ) = (cid:8) ( x, y ) ∈ R d × R d : x ∈ K, x + y ∈ K, (cid:104) y, v (cid:105) = 1 (cid:9) . The set S ( K ) is convex. Consider the set G = { S ( K ) : K ∈ F} ; this family iscontained in an affine subspace of dimension 2 d − (cid:104) y, v (cid:105) = 1.The hypothesis of the theorem implies that every 2 d or fewer elements of G intersect,so (cid:84) G is nonempty by Helly’s theorem. If ( x, y ) ∈ (cid:84) G , then x, x + y ∈ F for everyset F ∈ F , which shows that the v -width of (cid:84) F is at least 1. (cid:3) As before, substituting fractional or colorful versions of Helly’s theorem in theproof provides corresponding variants of Theorem 3.2. In fact, we can substituteHelly’s theorem with Kalai and Meshulam’s topological extension of the colorfulHelly theorem [25], which generalizes the coloring structure to an arbitrary matroid.Such an extension is much stronger than the v -width results obtained in [33]. Forlater reference, we explicitly state the fractional version of Theorem 3.2 derivedfrom the fractional Helly theorem (first proved by Katchalski and Liu [26]). Theorem 3.3.
For every positive integer d and real number α ∈ (0 , , there existsa real number β = β ( α, d ) > such that the following holds for every nonzero vector v ∈ R d . If (cid:84) H contains has v -width greater than or equal to 1 for at least α (cid:0) |F| d (cid:1) subcollections H ⊆ F of d sets, then there exists a subfamily G ⊆ F such that |G| ≥ β |F| whose intersection has v -width greater than or equal to 1. Even though the theorem above is already known [33], this method allows us todeduce the explicit bound β ( α, d ) ≥ − (1 − α ) / d by using the results of Kalai[23, 24]. We now use the Helly-type theorem for v -width to prove a colorful versionof Theorem 1.5. Theorem 3.4.
Let ρ be a Minkowski norm in R d whose unit ball is a polytope with k facets, and let F , . . . , F kd be finite families of convex sets in R d . If (cid:84) di =1 F i has ρ -diameter at least for every kd -tuple ( F i ) di =1 such that F i ∈ F i for each i , thenthere exists an index l ∈ [ kd ] such that (cid:84) F l has ρ -diameter at least . Moreover,the same statement is not true if kd is replaced by kd − .Proof. We prove the contrapositive. Assume that F , . . . , F kd are finite families ofconvex sets in R d such that (cid:84) F i has ρ -diameter at most 1 for each i ∈ [ kd ]. Wewant to find a colorful kd -tuple whose intersection has ρ -diameter at most 1.For each facet L j of P , let v j be the vector in R d such that (cid:104) x, v (cid:105) = 1 forevery x ∈ L j . We choose a labelling of the facets so that L ( k/ j = − L j for each j ∈ [ k/ ρ -width of F i and (3.1), the v j -width of (cid:84) F i isat most 1 for each i ∈ [ kd ] and j ∈ [ k/ d families F j − d +1 , . . . , F jd impliesthat there is a set F i +2( j − d ∈ F i +2( j − d for each i ∈ [2 d ] such that (cid:84) di =1 F j − d + i has v j -width less than or equal to 1.Let G denote the family { F , . . . , F kd } . By construction, G has exactly oneelement from each F i . Its intersection (cid:84) G has v j -width at most 1 for every j ∈ [ k/ ρ -diameter of (cid:84) G is at most 1 by (3.1). DILLON AND SOBER ´ON
Now we prove optimality. Consider a set { x , . . . , x k } of points in R d such that x i is in the relative interior of L i and x i = − x ( k/ i . For each i ∈ [ k/ d closed half-spaces such that • each half-space contains every point in { x j } j (cid:54) = i , • the intersection of the d half-spaces with L i is the singleton { x i } , and • the intersection of any d − y such that (cid:104) y, v i (cid:105) > F be the collection all kd half-spaces. The intersection (cid:84) F is contained in P , so its ρ -diameter is at most 2. For any subset F (cid:48) ⊆ F of size kd −
1, there is afacet L i of P with at most d − F (cid:48) . Therefore, thereexists a point ˜ x ∈ (cid:84) F (cid:48) outside of P such that the segment 0˜ x intersects L i . We canchoose ˜ x close enough to x i so that the segment between − x i and ˜ x has ρ -lengthgreater than 2. Therefore, the ρ -diameter (cid:84) F (cid:48) is greater than 2 for every subset F (cid:48) ⊆ F of size kd −
1, while the ρ -diameter of (cid:84) F is at most 2. Taking F i = F for each i ∈ [ kd −
1] shows the optimality of the parameter kd . (cid:3) Setting F i = F for each i ∈ [ kd ] in Theorem 3.4 proves the Helly-type state-ment in Theorem 1.5, and the proof the optimality of kd also carries over to themonochromatic version.Theorem 1.5 implies the following more general version of Corollary 1.8. Theorem 3.5.
Let p ≥ and F be a finite family of convex sets in R d . If theintersection of every d or fewer sets in F has (cid:96) p -diameter greater than or equalto 1, then (cid:84) F has (cid:96) p -diameter greater than or equal to d − /p .Proof. A set in R d with (cid:96) p -diameter at least 1 has (cid:96) ∞ -diameter at least d − /p . Sincethe unit ball in the (cid:96) ∞ norm is a polytope with 2 d facets, we can employ Theorem1.5 to conclude that the (cid:96) ∞ -diameter of (cid:84) F is at least d − /p . The (cid:96) p -diameter of F is at least d − /p as well. (cid:3) The (cid:96) norm is a useful lens with which to compare our results. Theorem 3.5 saysthat we can bound the (cid:96) -diameter of (cid:84) F by d − if we know that the intersection ofevery 2 d sets in F has (cid:96) -diameter greater than or equal to 1, whereas Theorem 1.5says that we can bound the (cid:96) -diameter of (cid:84) F by 1 if we know that the intersectionof every d d in F sets has (cid:96) -diameter greater than or equal to 1. Neither of theseconsequences implies the other.We now present two additional proofs of the Helly-type statement in Theorem1.5. The first proof uses the following lemma, in which the boundary of a set K ⊆ R d is denoted by ∂K . Lemma 3.6.
Let P ⊆ R d be a centrally symmetric polytope with k facets and G bea finite family of sets in R d such that (1) K ∩ ( R d × L ) is convex for every facet L of P and every K ∈ G , and (2) if x, y ∈ R d , K ∈ G , and ( x, y ) ∈ K , then ( x + y, − y ) ∈ K .If the intersection of every kd or fewer sets in G contains a point in R d × ∂P , then (cid:84) G contains a point in R d × ∂P .Proof. The general approach is similar to that of Radon’s proof of Helly’s theorem[29]. We proceed by induction on |G| . If |G| ≤ kd , there is nothing to prove, so weassume the result holds for for all collections of convex sets with n ≥ kd members.Let G = { K , . . . , K n +1 } be a collection of n +1 convex sets in R d that satisfies thehypothesis of the lemma. The induction hypothesis implies that for each i ∈ [ n + 1]there is a point ( x i , y i ) ∈ (cid:84) ( G \ K i ) such that y i ∈ ∂P . Grouping the facets of P byopposing pairs, there must be a pair of facets L, − L whose union contains at least n + 1 k/ > d M´ELANGE OF DIAMETER HELLY-TYPE THEOREMS 7 points in { y i } n +1 i =1 . By replacing ( x i , y i ) by ( x i + y i , − y i ) if necessary, the facet L contains at least 2 d + 1 points in { y i } n +1 i =1 . Therefore R d × L contains at least 2 d + 1points in { ( x i , y i ) } n +1 i =1 . Since R d × L is a (2 d − d + 1 points yields a partition of them into twosets A and B whose convex hulls intersect. Any point in conv( A ) ∩ conv( B ) is in (cid:84) G as well as R d × L ⊆ R d × ∂P . (cid:3) Second proof of Theorem 1.5.
Let P be the unit ball of ρ . For each convex set K ⊆ R d , let S ( K ) = (cid:8) ( x, y ) ∈ R d × R d : x ∈ K, x + y ∈ K, ρ ( y ) = 1 (cid:9) . The conditions of Theorem 1.5 ensure that we can apply Lemma 3.6 to the family G = { S ( K ) : K ∈ F} . Given a point ( x, y ) ∈ (cid:84) G ∩ ( R d × ∂P ), every set in F contains the segment between x and x + y ; since ρ ( y ) = 1, the ρ -diameter of (cid:84) G isat least 1. (cid:3) Our final proof of Theorem 1.5 relies on a limit argument.
Third proof of Theorem 1.5.
We may assume without loss of generality that everyset in F is compact. We denote by K n the n -fold product of the set K . Let f : ( R d ) k → R be defined by f ( x , y , . . . , x k/ , y k/ ) = k/ (cid:88) i =1 (cid:104) y i − x i , v i (cid:105) . If a set K ⊆ R d has ρ -diameter greater than or equal to 1, then it has v i -widthat least 1 for some i ∈ [ k/ x ∈ K k such that f (¯ x ) ≥
1. For each K ∈ F , consider the set S ( K ) = { ¯ x ∈ K k : f (¯ x ) = 1 } , which is convex and lies in an affine subspace of dimension kd −
1. Therefore,an application of Helly’s theorem implies that ( (cid:84) F ) k contains a point ¯ a with f (¯ a ) = 1.Now consider the function g : ( R d ) k → R defined by g ( x , y , . . . , x k/ , y k/ ) = max ≤ i ≤ k/ (cid:104) y i − x i , v i (cid:105) . If g (¯ a ) = 1, we are done. Otherwise, equation (3.1) implies that that for each kd -tuple F (cid:48) ⊆ F , there are two points x, y ∈ (cid:84) F (cid:48) and an index i ∈ [ k ] such that (cid:104) y − x, v i (cid:105) ≥
1. Replacing the corresponding coordinates of ¯ a by x, y , we obtain anew point ¯ x ∈ ( (cid:84) F (cid:48) ) k such that f (¯ x ) ≥ − g (¯ a )).Bootstrapping the previous arguments, we can find a point ¯ a ∈ ( (cid:84) F ) k suchthat f (¯ a ) = 2 − g (¯ a ). Iterating this argument creates a sequence (¯ a n ) ∞ n =1 in( (cid:84) F ) k such that f (¯ a n ) ≥ n − n − (cid:88) i =1 g (¯ a i )for each n ∈ N . Let β n = max { g (¯ a ) , . . . , g (¯ a n ) } . We have β n ≥ g (¯ a n ) ≥ f (¯ a n ) d ≥ n − (cid:80) n − i =1 g (¯ a i ) d ≥ n − ( n − β n d . In other words, the ρ -diameter of (cid:84) F is at least β n ≥ n/ ( d + n −
1) for every n ≥ n → ∞ finishes the proof. (cid:3) The next result shows that there is no exact Helly-type theorem for diameter forany norm whose unit ball is not a polytope.
DILLON AND SOBER ´ON
Theorem 3.7.
Let ρ be a Minkowski norm in R d whose unit ball is not a polytope.Then, for every integer n there exists a finite family G of convex sets such that theintersection of every n or fewer sets in G has ρ -diameter greater than or equal to , but the ρ -diameter of (cid:84) G is strictly less than .Proof. Let P be the unit ball of ρ and F be the infinite family of closed containment-minimal half-spaces that contain P . We parametrize F using the unit sphere S d − by associating each vector x ∈ S d − with the half-space H x ∈ F whose boundinghyperplane is perpendicular to it and that contains an infinite ray in the directionof x . For any finite family F (cid:48) ⊆ F , the unit ball P is strictly contained in (cid:84) F (cid:48) , sothe ρ -diameter of (cid:84) F (cid:48) is strictly larger than 2.We define a function f : ( S d − ) n → R by f ( x , . . . , x n ) = min (cid:26) , diam ρ (cid:16) n (cid:92) i =1 H x i (cid:17)(cid:27) . The minimum ensures that f is well-defined when (cid:84) ni =1 H x i is unbounded. Thefunction f is continuous, and f ( x , . . . , x n ) > n -tuple in ( S d − ) n . Sincethe domain of f is compact, f attains a minimum value s n > ε = ( s n − /
3. Standard results on approximation of convex sets by poly-topes show that there exists a polytope Q such that P ⊂ Q ⊂ (1 + ε ) P . Inparticular, diam ρ ( Q ) ≤ (1 + ε ) diam ρ ( P ) = 2(1 + ε ) < s n .We define G ⊆ F to be the family of half-spaces in F whose bounding hyperplanesare parallel to some facet of Q , scaled by a factor of 1 /s n . The intersection of every n or fewer sets in G has ρ -diameter greater than or equal to 1. But (cid:84) G ⊆ (1 /s n ) Q ,so its ρ -diameter is strictly less than 1. (cid:3) Diameter results for d -tuples We combine Theorem 3.2 with volume concentration properties of balls to provetheorems 1.2 and 1.3. The properties described below can be found in Keith Ball’sexpository notes [4].Let B be a ball centered at the origin, c >
0, and u be a unit vector. The c -cap of B in the direction u is C ( B, c, u ) = { x ∈ B : (cid:104) x, u (cid:105) ≥ c } . For two unit vectors u and v , we have that v ∈ C ( B, c, u ) if and only if u ∈ C ( B, c, v ).Let B d be the unit ball in R d , and let r d be the radius of a volume-one ball in R d . Asymptotically, r d ∼ d / / √ πe . For a fixed unit vector u and real number x , the ( d − r d B d with the hyperplane { y ∈ R d : (cid:104) u, y (cid:105) = x } converges to √ e exp( − πex ) as d → ∞ . In other words,the volume of the region { y ∈ R d : |(cid:104) y, u (cid:105)| < x } ∩ r d B d converges as d → ∞ , andconverges to zero as x →
0. For any fixed constant c , we define γ ( c ) = inf d ≥ vol (cid:20) C (cid:18) r d B d , cr d √ d , u (cid:19) ∪ C (cid:18) r d B d , cr d √ d , − u (cid:19)(cid:21) (4.1)= 1vol ( B d ) inf d ≥ vol (cid:20) C (cid:18) B d , c √ d , u (cid:19) ∪ C (cid:18) B d , c √ d , − u (cid:19)(cid:21) (4.2)This is the remaining volume of r d B d after removing a slab centered at the ori-gin with width approximately 2 c/ √ πe (see Figure 1). From the discussion above, γ ( c ) → c →
0. Equation (4.2) shows that γ ( c ) is the fraction of the vol-ume of two opposite ( cd − / )-caps in the unit sphere. If c < √
2, the volume of C (cid:0) r d B d , cr d d − / , u (cid:1) is strictly positive for each d ≥ d → ∞ . So γ ( c ) > c ∈ (0 , √ M´ELANGE OF DIAMETER HELLY-TYPE THEOREMS 9 x r d B d Figure 1.
The volume of the region in r d B d between two parallelhyperplanes at distance x from the origin converges as d → ∞ . Proof of Theorem 1.2.
Let n = |F| . For each subfamily F (cid:48) ⊆ F of 2 d sets whoseintersection has diameter greater than or equal to 1, assign a unit vector u F (cid:48) suchthat (cid:84) F (cid:48) contains a unit segment with direction u F (cid:48) . Let G be the collection ofsets C (cid:16) B d , cd − / , u F (cid:48) (cid:17) ∪ C (cid:16) B d , cd − / , − u F (cid:48) (cid:17) where F (cid:48) is a 2 d -tuple of F whose intersection has diameter greater than or equalto 1. Each such set covers at least a γ ( c ) fraction of the volume of the unit ball.Therefore, the total volume covered amongst all sets in G is at least γ ( c ) · α (cid:18) n d (cid:19) . Since (cid:83) G does not contain the origin, there is a nonzero point x in the unit ballcovered at least γ ( c ) · α (cid:0) n d (cid:1) times by G . Setting v = x/ || x || , the set C (cid:16) B d , cd − / , v (cid:17) ∪ C (cid:16) B d , cd − / , − v (cid:17) contains at least γ ( c ) · α (cid:0) n d (cid:1) different vectors u F (cid:48) . Thus, the intersection of at least γ ( c ) · α (cid:0) n d (cid:1) different 2 d -tuples of F have v -width greater than or equal to cd − / .An application of Theorem 3.3 finishes the proof, using the optimal bound for β due to Kalai [23, 24]. (cid:3) Proof of Theorem 1.3.
The proof is analogous to that of Theorem 1.2. In place ofthe fractional Helly theorem, we apply the colorful fractional theorem of B´ar´any,Fodor, Montejano, Oliveros, and P´or [6] with the bound for β by Kim [27]. (cid:3) Acknowlegments
The authors thank Silouanos Brazitikos and Gennadiy Averkov for their com-ments on this work.
References [1] Iskander Aliev, Robert Bassett, Jes´us A. De Loera, and Quentin Louveaux,
A quantitativeDoignon-Bell-Scarf theorem , Combinatorica (2016), no. 3, 313–332.[2] Nina Amenta, Jes´us A. De Loera, and Pablo Sober´on, Helly’s theorem: New variations andapplications , Algebraic and geometric methods in discrete mathematics, 2017, pp. 55–95.[3] Gennadiy Averkov, Bernardo Gonz´alez Merino, Ingo Paschke, Matthias Schymura, and StefanWeltge,
Tight bounds on discrete quantitative Helly numbers , Advances in Applied Mathe-matics (2017), 76–101.[4] Keith M. Ball, An elementary introduction to modern convex geometry , Flavors of geometry,1997, pp. 1–58.[5] Imre B´ar´any,
A generalization of Carath´eodory’s theorem , Discrete Mathematics (1982),no. 2-3, 141–152.[6] Imre B´ar´any, Ferenc Fodor, Luis Montejano, Deborah Oliveros, and Attila P´or, Colourful andfractional ( p, q ) -theorems , Discrete & Computational Geometry (2014), no. 3, 628–642.[7] Imre B´ar´any, Meir Katchalski, and J´anos Pach, Quantitative Helly-type theorems , Proceed-ings of the American Mathematical Society (1982), no. 1, 109–114.[8] Imre B´ar´any, Meir Katchalski, and J´anos Pach, Helly’s theorem with volumes , The AmericanMathematical Monthly (1984), no. 6, 362–365.[9] Imre B´ar´any and Jiˇr´ı Matouˇsek, A fractional Helly theorem for convex lattice sets , Advancesin Mathematics (2003), no. 2, 227–235.[10] David Bell,
A theorem concerning the integer lattice , Studies in Applied Mathematics (1976), no. 2, 187–188.[11] Silouanos Brazitikos, Brascamp–Lieb inequality and quantitative versions of Helly’s theorem ,Mathematika (2017), no. 1, 272–291.[12] Silouanos Brazitikos, Quantitative Helly-type theorem for the diameter of convex sets , Dis-crete & Computational Geometry (2017), no. 2, 494–505.[13] Silouanos Brazitikos, Polynomial estimates towards a sharp Helly-type theorem for the di-ameter of convex sets , Bulletin of the Hellenic mathematical society (2018), 19–25.[14] G´abor Dam´asdi, Vikt´oria F¨oldv´ari, and M´arton Nasz´odi, Colorful Helly-type theorems forellipsoids , 2019. arXiv:1909.04997v2 [math.MG].[15] Jes´us A. De Loera, Reuben N. La Haye, Deborah Oliveros, and Edgardo Rold´an-Pensado,
Helly numbers of algebraic subsets of R d and an extension of Doignon’s theorem , Advancesin Geometry (2017), no. 4, 473–482.[16] Jes´us A. De Loera, Reuben N. La Haye, David Rolnick, and Pablo Sober´on, Quantitativecombinatorial geometry for continuous parameters , Discrete & Computational Geometry (2017), no. 2, 318–334.[17] Jes´us A. De Loera, Reuben N. La Haye, David Rolnick, and Pablo Sober´on, QuantitativeTverberg theorems over lattices and other discrete sets , Discrete & Computational Geometry (2017), no. 2, 435–448.[18] Travis Dillon, Discrete quantitative Helly-type theorems with boxes , 2020. arXiv:2008.06013v1[math.CO].[19] Jean-Paul Doignon,
Convexity in cristallographical lattices , Journal of Geometry (1973),no. 1, 71–85.[20] Tom´as Fernandez Vidal, Daniel Galicer, and Mariano Merzbacher, Continuous quantitativeHelly-type results , 2020. arXiv:2006.09472v2 [math.MG].[21] Eduard Helly, ¨Uber Mengen konvexer K¨orper mit gemeinschaftlichen Punkte , Jahresberichtder Deutschen Mathematiker-Vereinigung (1923), 175–176.[22] Andreas Holmsen and Rephael Wenger, Helly-type theorems and geometric transversals ,Handbook of discrete and computational geometry, 2017, pp. 91–123.[23] Gil Kalai,
Characterization of f -vectors of families of convex sets in R d part I: Necessity ofEckhoff’s conditions , Israel Journal of Mathematics (1984), no. 2-3, 175–195.[24] Gil Kalai, Characterization of f -vectors of families of convex sets in R d part II: Sufficiencyof Eckhoff’s conditions , Journal of Combinatorial Theory, Series A (1986), no. 2, 167–188.[25] Gil Kalai and Roy Meshulam, A topological colorful Helly theorem , Advances in Mathematics (2005), no. 2, 305–311.[26] Meir Katchalski and Andy Liu,
A problem of geometry in R n , Proceedings of the AmericanMathematical Society (1979), no. 2, 284–288.[27] Minki Kim, A note on the colorful fractional Helly theorem , Discrete Mathematics (2017),no. 1, 3167–3170.[28] M´arton Nasz´odi,
Proof of a conjecture of B´ar´any, Katchalski and Pach , Discrete & Compu-tational Geometry (2016), no. 1, 243–248. M´ELANGE OF DIAMETER HELLY-TYPE THEOREMS 11 [29] Johann Radon,
Mengen konvexer K¨orper, die einen gemeinsamen Punkt enthalten , Mathe-matische Annalen (1921), no. 1, 113–115.[30] David Rolnick and Pablo Sober´on, Quantitative ( p, q ) theorems in combinatorial geometry ,Discrete Mathematics (2017), no. 10, 2516–2527.[31] Sherry Sarkar, Alexander Xue, and Pablo Sober´on, Quantitative combinatorial geometry forconcave functions , 2019. arXiv:1908.04438v1 [math.CO].[32] Herbert Scarf,
An observation on the structure of production sets with indivisibilities , Pro-ceedings of the National Academy of Sciences of the United States of America (1977),no. 9, 3637–3641.[33] Pablo Sober´on, Helly-type theorems for the diameter , Bulletin of the London MathematicalSociety (2016), no. 4, 577–588. Lawrence University, 711 E. Boldt Way, Appleton, WI 54911
E-mail address : [email protected] Baruch College, City University of New York, One Bernard Baruch Way, NewYork, NY 10010
E-mail address ::