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From Bulletin of the AMS

Five sets in $\mathbb{R}^3$ that pairwise intersect and have no line transversal.

Helly-type problems

by Imre Bárány and Gil Kalai

Abstract

In this paper we present a variety of problems in the interface between combinatorics and geometry around the theorems of Helly, Radon, Carathéodory, and Tverberg. Through these problems we describe the fascinating area of Helly-type theorems and explain some of their main themes and goals.

1. Helly, Carathéodory, and Radon theorems

In this paper, we present a variety of problems in the interface between combinatorics and geometry around the theorems of Helly, Radon, Carathéodory, and Tverberg.

Helly's theorem [Hel23] asserts that for a family $\{K_1,K_2,\ldots, K_n\}$ of convex sets in $\mathbb R^d$, where $n \ge d+1$, if every $d+1$ of the sets have a point in common, then all of the sets have a point in common. The closely related Carath\'eodory theorem [Car07] states that for $S \subset \mathbb R^d$, if $x \in \operatorname{conv\;} S$, then $x \in \operatorname{conv\;} R$ for some $R \subset S$, $|R| \le d+1$.

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