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

Abstract

In this note we study a conjecture by Jerónimo-Castro, Magazinov and Soberón which generalized a question posed by Dol'nikov. Let F1,F2,…,Fn 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, ⋃i≠jFi can be pierced by at most 4 points. To do so, we use previous ideas from Gomez-Navarro and Roldán-Pensado together with an approximation result closely tied to the Banach-Mazur distance to the square.