The Colored Hadwiger Transversal Theorem in $\mathbb R^d$

Abstract

Hadwiger's transversal theorem gives necessary and sufficient conditions for a family of convex sets in the plane to have a line transversal. A higher dimensional version was obtained by Goodman, Pollack and Wenger, and recently a colorful version appeared due to Arocha, Bracho and Montejano. We show that it is possible to combine both results to obtain a colored version of Hadwiger's theorem in higher dimensions. The proofs differ from the previous ones and use a variant of the Borsuk-Ulam theorem. To be precise, we prove the following. Let $F$ be a family of convex sets in $\mathbb R^d$ in bijection with a family $P$ of points in $\mathbb R^{d-1}$. Assume that there is a coloring of $F$ with sufficiently many colors such that any colorful Radon partition of points in $P$ corresponds to a colorful Radon partition of sets in $F$. Then some monochromatic subfamily of $F$ has a hyperplane transversal.