On a colorful problem by Dol'nikov concerning translates of convex bodies
Combinatorics
2023-09-26 v2
Abstract
In this note we study a conjecture by Jer\'onimo-Castro, Magazinov and Sober\'on which generalized a question posed by Dol'nikov. Let be families of translates of a convex compact set in the plane so that each two sets from distinct families intersect. We show that, for some , can be pierced by at most points. To do so, we use previous ideas from Gomez-Navarro and Rold\'an-Pensado together with an approximation result closely tied to the Banach-Mazur distance to the square.
Cite
@article{arxiv.2307.07714,
title = {On a colorful problem by Dol'nikov concerning translates of convex bodies},
author = {Leonardo Martínez-Sandoval and Edgardo Roldán-Pensado},
journal= {arXiv preprint arXiv:2307.07714},
year = {2023}
}