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arXiv · 2307.07714

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

Leonardo Martínez-Sandoval·Edgardo Roldán-Pensado
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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.

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Leonardo Martínez-Sandoval, Edgardo Roldán-Pensado. 2023-07-15. On a colorful problem by Dol'nikov concerning translates of convex bodies. https://arxiv.org/abs/2307.07714

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