A mélange of diameter Hellytype theorems
Abstract
A Hellytype 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 Hellytype 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 Hellytype theorems for the property of ``containing $k$ colinear integer points.
 Publication:

arXiv eprints
 Pub Date:
 August 2020
 arXiv:
 arXiv:2008.13737
 Bibcode:
 2020arXiv200813737D
 Keywords:

 Mathematics  Metric Geometry;
 Mathematics  Combinatorics
 EPrint:
 11 pages, 1 figure