Colorful Helly-type Theorems for the Volume of Intersections of Convex Bodies
CCOLORFUL HELLY-TYPE THEOREMS FOR THE VOLUME OFINTERSECTIONS OF CONVEX BODIES
G ´ABOR DAM ´ASDI, VIKT ´ORIA F ¨OLDV ´ARI, AND M ´ARTON NASZ ´ODI
Abstract.
We prove the following Helly-type result. Let C , . . . , C d be finite familiesof convex bodies in R d . Assume that for any colorful selection of 2 d sets, C i k ∈ C i k foreach 1 ≤ k ≤ d with 1 ≤ i < · · · < i d ≤ d , the intersection d (cid:84) k =1 C i k is of volume atleast 1. Then there is an 1 ≤ i ≤ d such that (cid:84) C ∈C i C is of volume at least d − O ( d ) . Introduction
According to Helly’s Theorem, if the intersection of any d +1 members of a finite familyof convex sets in R d is non-empty, then the intersection of all members of the family isnon-empty. A generalization of Helly’s Theorem, known as the
Colorful Helly Theorem , wasgiven by Lov´asz, and later by B´ar´any [B´ar82]: If C , . . . , C d +1 are finite families (colorclasses) of convex sets in R d , such that for any colorful selection C ∈ C , . . . , C d +1 ∈ C d +1 ,the intersection d +1 (cid:84) i =1 C i is non-empty, then for some j , the intersection (cid:84) C ∈C j C is also non-empty. Another variant of Helly’s Theorem was introduced by B´ar´any, Katchalski and Pach[BKP82], whose
Quantitative Volume Theorem states the following.
Assume thatthe intersection of any d members of a finite family of convex sets in R d is of volume atleast 1. Then the volume of the intersection of all members of the family is of volume atleast c d , a constant depending on d only. They proved that one can take c d = d − d and conjectured that it should hold with c d = d − cd for an absolute constant c >
0. It was confirmed with c d ≈ d − d in [Nas16], whoseargument was then refined by Brazitikos [Bra17], who showed that one may take c d ≈ d − d/ . For more on quantitative Helly-type results, see the surveys [HW18, DGMM19].In the present paper, we combine the two directions: colorful and quantitative.1.1. Ellipsoids and volume.
A well known consequence of John’s Theorem (Corol-lary 2.2), is that any compact convex set K with non-empty interior contains a uniqueellipsoid E of maximal volume, moreover, E enlarged around its center by a factor d con-tains K (cf. [Bal97]). It follows that the volume of the largest ellipsoid contained in K isof volume at least d − d Vol( K ). More precise bounds for this volume ratio are known (cf.[Bal97]), but we will not need them.As shown in [Nas16, Section 3], in the Quantitative Volume Theorem, the d − cd factoris sharp up to the absolute constant c . In particular, for every sufficiently large positiveinteger d , there is a family of convex sets satisfying the assumptions of the theorem whoseintersection is of volume roughly d − d/ .John’s Theorem and the fact above yield that bounding the volume of intersections andbounding the volume of ellipsoids contained in the intersections are essentially equivalent a r X i v : . [ m a t h . M G ] J u l roblems: the only difference is a multiplicative factor d d which is of no consequence,unless one wants to find the best constants in the exponent. Thus, from this point on,we phrase our results in terms of the volume of ellipsoids contained in intersections. Itsbenefit is that this is how in the proofs we actually “find volume”: we find ellipsoids oflarge volume.1.2. Main result: few color classes.
Our main result is the following.
Theorem 1.1 (Colorful Quantitative Volume Theorem with Ellipsoids – Few ColorClasses) . Let C , . . . , C d be finite families of convex bodies in R d . Assume that for anycolorful selection of d sets, C i k ∈ C i k for each ≤ k ≤ d with ≤ i < · · · < i d ≤ d ,the intersection d (cid:84) k =1 C i k contains an ellipsoid of volume at least 1. Then, there exists an ≤ i ≤ d such that (cid:84) C ∈C i C contains an ellipsoid of volume at least c d d − d / with anabsolute constant c ≥ . We rephrase this theorem in terms of the volume of intersections, as this form may bemore easily applicable.
Corollary 1.2 (Colorful Quantitative Volume Theorem – Few Color Classes) . Let C ,. . . , C d be finite families of convex bodies in R d . Assume that for any colorful selectionof d sets, C i k ∈ C i k for each ≤ k ≤ d with ≤ i < · · · < i d ≤ d , the intersection d (cid:84) k =1 C i k is of volume at least 1.Then, there exists an ≤ i ≤ d such that Vol (cid:18) (cid:84) C ∈C i C (cid:19) ≥ c d d − d / with an absoluteconstant c ≥ . Observe that the smaller the number of color classes in a colorful Helly-type theorem,the stronger the theorem is. For example, the Colorful Helly Theorem (see the top of thesection) is stated with d + 1 color classes, but it is easy to see that it implies the sameresult with (cid:96) ≥ d + 2 color classes, as the last (cid:96) − ( d + 1) color classes make the assumptionof the theorem stronger and the conclusion weaker. We note also that the Colorful HellyTheorem does not hold with less than d + 1 color classes, as the number d + 1 cannot bereplaced by any smaller number in Helly’s Theorem.The novelty of the proof of Theorem 1.1 is the following. As we will see later, similarlooking statements can be obtained by taking the Quantitative Volume Theorem as a“basic” Helly-type theorem, and combining it with John’s Theorem and a combinatorialargument. This approach yields results with d ( d + 3) / K to otherellipsoids contained in K .We find it an intriguing question whether one can decrease the number of color classesto 2 d (possibly with an even weaker bound on the volume of the ellipsoid obtained), andwhether an order d − cd lower bound on the volume of the ellipsoid can be shown.1.3. Earlier results and simple observations.
In 1937, Behrend [Beh37] (see alsoSection 6.17 of the survey [DGK63] by Danzer, Gr¨unbaum and Klee) proved a planarquantitative Helly-type result:
If the intersection of any 5 members of a finite family ofconvex sets in R contains an ellipse of area 1, then the intersection of all members of he family contains an ellipse of area 1 . We note that, since every convex set in R isthe intersection of the half-planes containing it, the result is equivalent to the formallyweaker statement where the family consists of half-planes only. This is the form in whichit is stated in [DGK63].In [DGK63, Section 6.17], it is mentioned that John’s Theorem (Theorem 2.1) should beapplicable to extend Behrend’s result to higher dimensions. We spell out this argument,and present a straightforward proof of the following. Proposition 1.3 (Helly-type Theorem with Ellipsoids) . Let C be a finite family of atleast d ( d +3) / convex sets in R d , and assume that for any selection C , . . . , C d ( d +3) / ∈ C ,the intersection d ( d +3) / (cid:84) i =1 C i contains an ellipsoid of volume 1. Then (cid:84) C ∈C C also containsan ellipsoid of volume 1. The number d ( d + 3) / d , there existsa family of d ( d + 3) / B d is the maximum volumeellipsoid contained in their intersection, but B d is not the maximum volume ellipsoidcontained in the intersection of any proper subfamily of them. That is, the intersectionof any subfamily of d ( d + 3) / − B d (which we denote by ω d = Vol( B d )), and yet, the intersection of allmembers of the family does not contain an ellipsoid of larger volume than ω d . This followsfrom the much stronger result, Theorem 4 in [Gru88] by Gruber.We prove a colorful version of Proposition 1.3. Proposition 1.4 (Colorful Quantitative Volume Theorem with Ellipsoids – Many ColorClasses) . Let C , . . . , C d ( d +3) / be finite families of convex bodies in R d , and assume thatfor any colorful selection C ∈ C , . . . , C d ( d +3) / ∈ C d ( d +3) / , the intersection d ( d +3) / (cid:84) i =1 C i contains an ellipsoid of volume 1. Then for some j , the intersection (cid:84) C ∈C j C contains anellipsoid of volume 1. According to [DLOR17] by De Loera et al., a monochromatic Helly-type theoremimplies a colorful version in a certain combinatorial setting . The novelty of Proposition 1.4is the geometric part of the proof, where we introduce an ordering on the family ofellipsoids, and study the properties of this ordering (see Section 2.3).Sarkar, Xue and Sober´on [SXS19, Corollary 1.0.5], using matroids, recently obtaineda result involving d ( d + 3) / d . Proposition 1.5 (Sarkar, Xue and Sober´on [SXS19]) . Let C , . . . , C d ( d +3) / be finite fami-lies of convex bodies in R d . Assume that for any colorful selection of d sets, C i k ∈ C i k foreach ≤ k ≤ d with ≤ i < · · · < i d ≤ d ( d + 3) / , the intersection d (cid:84) k =1 C i k contains anellipsoid of volume at least 1. Then, there exists an ≤ i ≤ d ( d + 3) / such that (cid:84) C ∈C i C has volume at least d − O ( d ) . For completeness, in Section 3.3, we sketch a brief argument showing that Propo-sition 1.5 immediately follows from our Proposition 1.4 and the Quantitative VolumeTheorem.The structure of the paper is the following. In Section 2, we introduce some preliminaryfacts and definitions, notably, an ordering on the family of ellipsoids of volume at least 1that are contained in a convex body. Section 3 contains the proofs of our results. . Preliminaries
John’s ellipsoid.Theorem 2.1 (John [Joh48]) . Let K ⊂ R d be a convex body. Then K contains a uniqueellipsoid of maximal volume. This ellipsoid is B d if and only if B d ⊂ K and thereare contact points u , . . . , u m ∈ bd ( K ) ∩ bd ( B d ) and positive numbers λ , . . . , λ m with d + 1 ≤ m ≤ d ( d +3)2 such that m (cid:88) i =1 λ i u i = 0 , and I d = m (cid:88) i =1 λ i u i u Ti , where I d denotes the d × d identity matrix and the u i are column vectors. The following is a well known corollary, see [Bal97, Lecture 3].
Corollary 2.2.
Assume that B d is the unique maximal volume ellipsoid contained in aconvex body K in R d . Then d B d ⊇ K . Colorful Helly Theorem.
We recall the Colorful Helly Theorem, as one of itsstraightforward corollaries will be used.
Theorem 2.3 (Colorful Helly Theorem, Lov´asz, B´ar´any [B´ar82]) . Let C , . . . , C d +1 befinite families of convex bodies in R d , and assume that for any colorful selection C ∈C , . . . , C d +1 ∈ C d +1 , the intersection d +1 (cid:84) i =1 C i is non-empty. Then for some j , the intersec-tion (cid:84) C ∈C j C is also non-empty. Corollary 2.4.
Let C , . . . , C d +1 be finite families of convex bodies, and L a convex bodyin R d . Assume that for any colorful selection C ∈ C , . . . , C d +1 ∈ C d +1 , the intersection d +1 (cid:84) i =1 C i contains a translate of L . Then for some j , the intersection (cid:84) C ∈C j C contains atranslate of L .Proof of Corollary 2.4. We use the following operation, the Minkowski difference of twoconvex sets A and B : A ∼ B := (cid:92) b ∈ B ( A − b ) . It is easy to see that A ∼ B is the set of those vectors t such that B + t ⊆ A .By the assumption, for any colorful selection C ∈ C , . . . , C d +1 ∈ C d +1 , we have d +1 (cid:84) i =1 ( C i ∼ L ) (cid:54) = ∅ . By Theorem 2.3, for some j , we have (cid:84) C ∈C j ( C ∼ L ) (cid:54) = ∅ , and thus, (cid:84) C ∈C j C contains a translate of L . (cid:3) Lowest ellipsoid.
We will follow Lov´asz’ idea of the proof of the Colorful HellyTheorem. The first step is to fix an ordering of the objects of study. This time, we arelooking for an ellipsoid and not a point in the intersection, therefore we need an orderingon the ellipsoids .For an ellipsoid E , we define its height as the largest value of the orthogonal projectionof E on the last coordinate axis, that is, max { x T e d | x ∈ E } , where e d = (0 , , . . . , , T . emma 2.5. Let C be a convex body that contains an ellipsoid of volume ω d := Vol( B d ) .Then there is a unique ellipsoid of volume ω d such that every other ellipsoid of volume ω d in C has larger height. Furthermore, if τ ∈ R denotes the height of this ellipsoid, thenthe largest volume ellipsoid of the convex body H τ ∩ C is this ellipsoid, where H τ denotesthe closed half-space H τ = { x ∈ R d | x T e d ≤ τ } . We call this ellipsoid the lowest ellipsoid in C . Proof of Lemma 2.5.
It is not difficult to see that H τ ∩ C does not contain any ellipsoidof volume larger than ω d . Indeed, otherwise for a sufficiently small (cid:15) >
0, the set H τ − (cid:15) ∩ C would contain an ellipsoid of volume equal to ω d , where H τ − (cid:15) denotes the closed half-space H τ − (cid:15) = { x ∈ R d | x T e d ≤ τ − (cid:15) } .Thus, by Theorem 2.1, B d is the unique largest volume ellipsoid of H τ ∩ C . It followsthat B d is the unique lowest ellipsoid of C . (cid:3) Quantitative Volume Theorem with Ellipsoids.
We will rely on the followingquantitative Helly theorem.
Theorem 2.6 (Quantitative Volume Theorem) . Let C , . . . , C n be convex sets in R d . As-sume that the intersection of any d of them is of volume at least 1. Then Vol (cid:18) n (cid:84) i =1 C i (cid:19) ≥ c d d − d/ with an absolute constant c > . As noted in Section 1, it is shown in [Nas16] that the d − d/ term cannot be improvedfurther than d − d/ . Corollary 2.7 (Quantitative Volume Theorem with Ellipsoids) . Let C , . . . , C n be convexsets in R d . Assume that the intersection of any d of them contains an ellipsoid of volumeat least 1. Then n (cid:84) i =1 C i contains an ellipsoid of volume at least c d d − d/ with an absoluteconstant c > . Theorem 2.6 was proved by B´ar´any, Katchalski and Pach [BKP82] with the weakervolume bound d − d . In [Nas16], the volume bound c d d − d was shown, and this argumentwas later refined by Brazitikos [Bra17] to obtain the bound presented above. An inspec-tion of the argument in [Nas16] shows that Corollary 2.7 holds with the slightly strongerbound c d d − d/ as well. However, as this constant in the exponent is of no consequence,we instead deduce Corollary 2.7 in the form presented above from Theorem 2.6. Proof of Corollary 2.7.
Let C , . . . , C n be convex sets in R d satisfying the assumptionsof Corollary 2.7. In particular, they satisfy the assumptions of Theorem 2.6, and hence,Vol (cid:18) n (cid:84) i =1 C i (cid:19) ≥ c d d − d/ . Finally, Corollary 2.2 yields that n (cid:84) i =1 C i contains an ellipsoid ofvolume at least c d d − d/ completing the proof of Corollary 2.7. (cid:3) Proofs
Proof of Proposition 1.3.
We will prove the following statement, which is clearlyequivalent to Proposition 1.3.
Assume that the largest volume ellipsoid contained in (cid:84) C ∈C C is of volume ω d = Vol( B d ) .Then there are d ( d + 3) / sets in C such that the largest volume ellipsoid in their inter-section is of volume ω d . he problem is clearly affine invariant, and thus, we may assume that the largestvolume ellipsoid in (cid:84) C ∈C C is the unit ball B d .By one direction of Theorem 2.1, there are contact points u , . . . , u m ∈ bd ( (cid:84) C ∈C C ) ∩ bd ( B d ) and positive numbers λ , . . . , λ m with d + 1 ≤ m ≤ d ( d +3)2 satisfying the equationsin Theorem 2.1. We can choose C , . . . C m ∈ C such that u i ∈ bd ( C i ) for i = 1 , . . . , m .By the other direction of Theorem 2.1, B d is the largest volume ellipsoid of m (cid:84) i =1 C i ,completing the proof of Proposition 1.3.3.2. Proof of Proposition 1.4.Lemma 3.1.
Let C , . . . , C d ( d +3) / be convex bodies in R d . Assume that K := d ( d +3) / (cid:84) i =1 C i contains an ellipsoid of volume ω d . Set K j := d ( d +3) / (cid:84) i =1 ,i (cid:54) = j C i , and let E denote the lowestellipsoid in K . Then there exists a j such that E is also the lowest ellipsoid of K j .Proof of Lemma 3.1. Let τ denote the height of E . By Lemma 2.5, E is the largest volumeellipsoid of K ∩ H τ , where H τ is the half-space defined in Lemma 2.5.Suppose that E is not the lowest ellipsoid in K j for every j ∈ { , . . . , d ( d + 3) / } . Since E ⊂ K ⊂ K j , this means that each K j contains a lower ellipsoid than E of volume ω d .Therefore we can choose a small (cid:15) > K j ∩ H τ − (cid:15) contains an ellipsoid of volume ω d for each j , where H τ − (cid:15) denotes the closed half-space H τ − (cid:15) = { x ∈ R d | x T e d ≤ τ − (cid:15) } .Let us consider now the following d ( d +3)2 + 1 sets: K , K , . . . , K d ( d +3) / , H τ − (cid:15) . If we takethe intersection of d ( d +3)2 of these sets, we obtain either K , or K j ∩ H τ − (cid:15) for some j . Byour assumption, K contains an ellipsoid of volume ω d . By the choice of (cid:15) , we have that K j ∩ H τ − (cid:15) also contains an ellipsoid of volume ω d . Hence, we can apply Proposition 1.3,which yields that C ∩· · ·∩ C d ( d +3) / ∩ H τ − (cid:15) = K ∩ H τ − (cid:15) also contains an ellipsoid of volume ω d . This contradicts the fact that E is the lowest ellipsoid in K , and thus, Lemma 3.1follows. (cid:3) We will prove the following statement, which is clearly equivalent to Proposition 1.4.
Assume that for every colorful selection C ∈ C , . . . , C d ( d +3) / ∈ C d ( d +3) / , the inter-section d ( d +3) / (cid:84) i =1 C i contains an ellipsoid of volume ω d . We will show that for some j , theintersection (cid:84) C ∈C j C contains an ellipsoid of volume ω d . By Lemma 2.5, we can choose the lowest ellipsoid in each of these intersections. Let usdenote the set of these ellipsoids as B . Since we have finitely many intersections, there isa highest one among these ellipsoids. Let us denote this ellipsoid by E max . E max is defined by some C ∈ C , . . . , C d ( d +3) / ∈ C d ( d +3) / . Once again let K j = d ( d +3) / (cid:84) i =1 ,i (cid:54) = j C i and K = d ( d +3) / (cid:84) i =1 C i . By Lemma 3.1, there is a j such that E max is the lowestellipsoid in K j . We will show that E max lies in every element of C j for this j .Fix a member C of C j . Suppose that E max (cid:54)⊂ C . Then E max (cid:54)⊂ C ∩ K j . By theassumption of Proposition 1.4, C ∩ K j contains an ellipsoid of volume ω d , since it is theintersection of a colorful selection of sets. Since C ∩ K j ⊂ K j , the lowest ellipsoid of C ∩ K j is at least as high as the lowest ellipsoid of K j . But the unique lowest ellipsoidof K j is E max , and E max (cid:54)⊂ C ∩ K j . So the lowest ellipsoid of C ∩ K j lies higher than max . This contradicts that E max was chosen to be the highest among the ellipsoids in B . So E max ⊂ C . Since C ∈ C j was chosen arbitrarily, we obtain that E max ⊂ (cid:84) C ∈C j C ,completing the proof of Proposition 1.4.3.3. Proof of Proposition 1.5.
Consider an arbitrary colorful selection of d ( d + 3) / c d d − d/ . It follows immediately from Proposition 1.4, that the intersection of oneof the color classes contains an ellipsoid of volume at least c d d − d/ , completing the proofof Proposition 1.5.3.4. Proof of Theorem 1.1.
We will prove the following statement, which is clearlyequivalent to Theorem 1.1.
Assume that the intersection of all colorful selections of d sets contains an ellipsoidof volume at least ω d . Then, there is an ≤ i ≤ d such that (cid:84) C ∈C i C contains an ellipsoidof volume at least c d d − d / ω d with an absolute constant c ≥ . Lemma 3.2.
Assume that B d is the largest volume ellipsoid contained in the convex set C in R d . Let E be another ellipsoid in C of volume at least δω d with < δ < . Thenthere is a translate of δd d − B d which is contained in E .Proof of Lemma 3.2. If the length of all d semi-axes a , . . . , a d of E are at least λ for some λ >
0, then clearly, λ B d + c ⊂ E , where c denotes the center of E . We will show that allthe semi-axes are long enough.By Corollary 2.2, E ⊂ C ⊂ d B d . Therefore, a i ≤ d for every i = 1 , . . . , d . Since thevolume of E is a · · · a d ω d ≥ δω d , we have a i ≥ δd d − for every i = 1 , . . . , d , completing theproof of Lemma 3.2. (cid:3) Consider the lowest ellipsoid in the intersection of all colorful selections of 2 d − B d . By possibly changing theindices of the families, we may assume that the selection is C ∈ C , . . . , C d − ∈ C d − .We call C d , C d +1 , . . . , C d the remaining families .Consider the half-space H = { x ∈ R d | x T e d ≤ } ⊃ B d . By Lemma 2.5, B d is thelargest volume ellipsoid contained in M := C ∩ · · · ∩ C d − ∩ H .Next, take an arbitrary colorful selection C d ∈ C d , C d +1 ∈ C d +1 , . . . , C d ∈ C d of theremaining d + 1 families. We claim that the intersection of any 2 d sets of C , . . . , C d − , H , C d , . . . , C d contains an ellipsoid of volume at least ω d . Indeed, if H is not among those 2 d sets, thenour assumption ensures this. If H is among them, then by the choice of H , the claimholds.Therefore, by Theorem 2.7, the intersection d (cid:92) i =1 C i ∩ H contains an ellipsoid E of volume at least δω d , where δ := c d d − d/ . Clearly, E ⊂ M .Since B d is the maximum volume ellipsoid contained in M , by Lemma 3.2, we havethat there is a translate of δd d − B d which is contained in E and thus in d (cid:84) i =2 d C i . hus, we have shown that any colorful selection C d ∈ C d , C d +1 ∈ C d +1 , . . . , C d ∈ C d of the remaining d + 1 families, d (cid:84) i =2 d C i contains a translate of the same convex body c d d − d/ B d . It follows from Corollary 2.4 that there is an index 2 d ≤ i ≤ d such that (cid:84) C ∈C i C contains a translate of c d d − d/ B d , which is an ellipsoid of volume c d d − d / ω d ,finishing the proof of Theorem 1.1.3.5. Proof of Corollary 1.2.
By Corollary 2.2, the volume of the largest ellipsoid ina convex body is at least d − d times the volume of the body. Corollary 1.2 now followsimmediately from Theorem 1.1. Acknowledgement.
Some of the results were part of GD’s master’s thesis. Part ofthe research was carried out while MN was a member of J´anos Pach’s chair of DCG atEPFL, Lausanne, which was supported by Swiss National Science Foundation Grants200020-162884 and 200021-165977. GD and MN also acknowledge the support of theNational Research, Development and Innovation Fund grant K119670 as well as that ofthe MTA-ELTE Lend¨ulet Combinatorial Geometry Research Group.
References [Bal97] Keith Ball. An elementary introduction to modern convex geometry. In
Flavors of geometry ,volume 31 of
Math. Sci. Res. Inst. Publ. , pages 1–58. Cambridge Univ. Press, Cambridge,1997. doi:10.2977/prims/1195164788.[B´ar82] Imre B´ar´any. A generalization of Carath´eodory’s theorem.
Discrete Math. , 40(2-3):141–152,1982. doi:10.1016/0012-365X(82)90115-7.[Beh37] Felix Behrend. ¨Uber einige Affininvarianten konvexer Bereiche.
Math. Ann. , 113(1):713–747,1937. doi:10.1007/BF01571662.[BKP82] Imre B´ar´any, Meir Katchalski, and J´anos Pach. Quantitative Helly-type theorems.
Proc.Amer. Math. Soc. , 86(1):109–114, 1982. doi:10.2307/2044407.[Bra17] Silouanos Brazitikos. Brascamp-Lieb inequality and quantitative versions of Helly’s theorem.
Mathematika , 63(1):272–291, 2017. doi:10.1112/S0025579316000255.[DGK63] Ludwig Danzer, Branko Gr¨unbaum, and Victor Klee. Helly’s theorem and its relatives. In
Proc. Sympos. Pure Math., Vol. VII , pages 101–180. Amer. Math. Soc., Providence, R.I.,1963.[DGMM19] Jes´us A. De Loera, Xavier Goaoc, Fr´ed´eric Meunier, and Nabil H. Mustafa. The discrete yetubiquitous theorems of Carath´eodory, Helly, Sperner, Tucker, and Tverberg.
Bull. Amer.Math. Soc. (N.S.) , 56(3):415–511, 2019. doi:10.1090/bull/1653.[DLOR17] 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. Adv. Geom. ,17(4):473–482, 2017. doi:10.1515/advgeom-2017-0028.[Gru88] Peter M. Gruber. Minimal ellipsoids and their duals.
Rend. Circ. Mat. Palermo (2) , 37(1):35–64, 1988. doi:10.1007/BF02844267.[HW18] Andreas Holmsen and Rephael Wenger. Helly-type theorems and geometric transversals. InJacob E. Goodman, Joseph O’Rourke, and Csaba D. T´oth, editors,
Handbook of discrete andcomputational geometry , pages 91–123. CRC Press, Boca Raton, FL, 2018. Third edition.[Joh48] Fritz John. Extremum problems with inequalities as subsidiary conditions. In
Studies andEssays Presented to R. Courant on his 60th Birthday, January 8, 1948 , pages 187–204.Interscience Publishers, Inc., New York, N. Y., 1948.[Nas16] M´arton Nasz´odi. Proof of a conjecture of B´ar´any, Katchalski and Pach.
Discrete Comput.Geom. , 55(1):243–248, 2016. doi:10.1007/s00454-015-9753-3.[SXS19] Sherry Sarkar, Alexander Xue, and Pablo Sober´on. Quantitative combinatorial geometry forconcave functions. arXiv e-prints , page arXiv:1908.04438, Aug 2019, 1908.04438. . D.: Dept. of Computer Science, ELTE E¨otv¨os Lor´and University, Budapest, Hun-gary E-mail address : [email protected] V. F.: Dept. of Geometry, ELTE E¨otv¨os Lor´and University, Budapest, Hungary
E-mail address : [email protected] M. N.: Alfr´ed R´enyi Inst. of Math.; MTA-ELTE Lend¨ulet Combinatorial GeometryResearch Group; Dept. of Geometry, ELTE E¨otv¨os Lor´and University, Budapest, Hun-gary
E-mail address : [email protected]@math.elte.hu