Very Colorful Theorems

Abstract

We prove several colorful generalizations of classical theorems in discrete geometry. Moreover, the colorful generalization of Kirchberger's theorem gives a generalization of the theorem of Tverberg on non-separated partitions.

The discrete yet ubiquitous theorems of Carathéodory, Helly, Sperner, Tucker, and Tverberg

“…There are examples showing that it is not sufficient that the convex hulls of triples contain the origin, but weaker relaxations are possible [274]. Arocha et al [29] also proved another "very colorful Carathéodory theorem".…”

Colorful Simplicial Depth, Minkowski Sums, and Generalized Gale Transforms

“…For the Colorful Carathéodory Theorem, the assumption of centeredness can be replaced by a weaker assumption (see, for example, [2,3,13,20]). This prompts the following question.…”

“…, C d } is called a colorful configuration. We call C centered if 0 ∈ relint conv(C i ) for all 0 ≤ i ≤ d. The assumption that C is centered can be considerably weakened (see, for example, [2,3,13,20]) but in this paper we will only consider centered configurations; see Question 1. We call a subset S ⊆ i C i a colorful simplex if |S ∩ C i | ≤ 1 for all i and we call the simplex S hitting if dim S = |S| − 1 = d and 0 ∈ conv(S).…”

Carathéodory-type Theorems à la Bárány

“…More recently, two independent groups have observed that these hypotheses can be weakened a bit more; namely, it suffices that the union of pairs of colours captures the point: Theorem 4 (Arocha et al [1], Holmsen et al [11]…”