On a colorful problem by Dol'nikov concerning translates of convex bodies

TL;DR

This paper investigates a geometric conjecture involving families of convex set translates in the plane, demonstrating that under certain intersection conditions, the union of all but one family can be pierced by at most four points, using approximation techniques related to Banach-Mazur distance.

Contribution

It proves a conjecture on intersecting translates of convex bodies, extending previous work with new approximation methods linked to Banach-Mazur distance.

Findings

Union of all but one family can be pierced by at most 4 points

Uses approximation tied to Banach-Mazur distance to the square

Builds on ideas from Gomez-Navarro and Roldán-Pensado

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 $F_1,F_2,\dots,F_n$ be families of translates of a convex compact set $K$ in the plane so that each two sets from distinct families intersect. We show that, for some $j$, $\bigcup_{i\neq j}F_i$ can be pierced by at most $4$ 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.