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arXiv · 2008.13737

A m\'elange of diameter Helly-type theorems

Travis Dillon·Pablo Soberón
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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.

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Travis Dillon, Pablo Soberón. 2020-08-31. A m\'elange of diameter Helly-type theorems. https://arxiv.org/abs/2008.13737

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