A mélange of diameter Helly-type theorems

Abstract

A Helly-type theorem for diameter provides a bound on the diameter of the intersection of a finite family of convex sets in ${\mathbb{R}}^d$ given some information on the diameter of the intersection of all sufficiently small subfamilies. We prove fractional and colorful versions of a longstanding conjecture by Bárány, Katchalski, and Pach. We also show that a Minkowski norm admits an exact Helly-type theorem for diameter if and only if its unit ball is a polytope and prove a colorful version for those that do. Finally, we prove Helly-type theorems for the property of ``containing $k$ colinear integer points.

11 pages, 1 figure