A m\'elange of diameter Helly-type theorems

TL;DR

This paper extends Helly-type theorems for diameter in convex geometry, proving fractional, colorful, and exact versions, and explores conditions for Minkowski norms and integer point containment.

Contribution

It introduces fractional and colorful variants of a longstanding diameter Helly conjecture and characterizes Minkowski norms with exact diameter Helly theorems.

Findings

Proved fractional and colorful diameter Helly theorems.

Characterized Minkowski norms with exact diameter Helly properties.

Established Helly-type theorems for colinear integer points.

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 B\'ar\'any, 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.