Colorful theorems for strong convexity

Abstract

Abstract. We prove two colorful Carathéodory theorems for strongly convex hulls, generalizing the colorful Caratéodory theorem for ordinary convexity by Imre Bárány, the non-colorful Carathéodory theorem for strongly convex hulls by the second author, and the "very colorful theorems" by the first author and others. We also investigate if the assumption of a "generating convex set" is really needed in such results and try to give a topological criterion for one convex body to be a Minkowski summand of another. IntroductionThe colorful Carathéodory theorem, discovered by Imre Bárány [3] (and independently in a dual form by Lovász), states that if X 0 , X 1 , . . . , X n are subsets in R n , each containing the origin in their convex hulls, then there exists a system of representatives x 0 ∈ X 0 , x 1 ∈ X 1 , . . . , x n ∈ X n such that the origin is contained in the convex hull of {x 0 , x 1 , . . . , x n }.The classical Carathéodory theorem [6] is recovered by setting X 0 = X 1 = · · · = X n . There are several remarkable applications of the colorful Carathéodory theorem in discrete geometry, such as in Sarkaria's proof of Tverberg's theorem and in the proof of the existence of weak ε-nets for convex sets. For further applications and references we recommend the reader to see Chapters 8, 9, and 10 in [19]. We recommend the survey [9] for general background on Carathéodory's theorem and its relatives.Recently there have been numerous generalizations of classical results from combinatorial convexity which focus on replacing convex sets by more general subsets of R n which are subject to certain topological constraints (see for instance [7,10,15,16,21] and references therein). These generalizations usually require tools from algebraic topology. In this paper we also use such topological tools, but we do so in order to solve problems which are purely affine.Our goal in this paper is to give an extension of the colorful Carathéodory theorem [3] to the notion of strong convexity and strongly convex hulls. This also generalizes the (noncolorful) Carathéodory theorem for strong convexity from [17]. In fact, the proof presented here is very similar to that in [17] In [18] it was shown that it is sufficient to test this property for those T that contain two elements; but we do not need this simplification here. It is relatively easy to check that all two-dimensional convex bodies, Euclidean balls, and simplices in every dimension are generating sets (see for instance section 3.2 of [23]). This property is also inherited under the Cartesian product operation. In [14] a criterion for checking this property for C 2 -smooth bodies was given, proving, in particular, that by sufficiently smooth perturbations of a ball one can obtain centrally symmetric generating sets which are not ellipsoids.in the above notation. The minimal K-strongly convex set containing a given set X is called its strongly convex hull, and can be found by the following formulaNote that the K-strongly convex hull is only defined for those X that are contain...

On the Carathéodory Number for Strong Convexity

“…The proof in [12,10] was exploiting the topological Helly property for the family of sets {F x } with Helly number n + 1, which we need to reduce now to n. In order to reduce the Helly number we assume without loss of generality 0 ∈ int K, take a slightly inflated K ε = (1 + ε)K and consider G x = (K − x) ∩ ∂K ε instead of F x trying to show the following for sufficiently small ε > 0:…”

“…In the works [14,1] (a recent English reference is [10]) a strengthening of the notion of convexity was studied. The first natural example is to call a convex body A ⊂ R n R-strongly convex for a positive real R, if A is an intersection of balls of radius R. This notion seems to have been rediscovered many times (see the references in the cited works), and in [14,1] it was generalized to the following: For a fixed convex body K ⊂ R n , another convex body A is K-strongly convex if it is an intersection of translates of K.…”

“…In [12] (as explained in [10] in English) it was shown that of K is a generating set then a version of Carathéodory theorem ( [6], see also the review [8]) for strong convexity holds for finite point sets X ⊂ R n :…”

“…As for the upper bounds, the example in [10,Section 5] shows that in dimensions n ≥ 3 there is no upper bound for the Carathéodory number for strong convexity, while in dimensions n ≤ 2 all convex bodies are generating sets.…”

“…Below we give proofs of the theorems stated here. In Section 6 we give a simple example of K with an infinite Carathéodory number for strong convexity, compared to the example in [10,Section 5]. Then in Section 7 we consider a related question of topological criteria for a convex compactum A to be a Minkowski summand of a convex body B, giving a version of the corresponding result of [10, Section 6].…”