A colorful Goodman-Pollack-Wenger theorem

TL;DR

This paper extends the Goodman-Pollack-Wenger theorem to a colorful setting, providing a topological proof that confirms a conjecture about hyperplane transversals in higher dimensions.

Contribution

It introduces a colorful generalization of the theorem and proves it using topological methods, specifically the Borsuk-Ulam theorem.

Findings

Confirmed a conjecture by Arocha, Bracho, and Montejano

Established a topological proof for the colorful theorem

Extended the theorem to hyperplane transversals in higher dimensions

Abstract

Hadwiger's transversal theorem gives necessary and sufficient conditions for the existence of a line transversal to a family of pairwise disjoint convex sets in the plane. These conditions were subsequently generalized to hyperplane transversals to general families of convex sets in $\mathbb{R}^d$ by Goodman, Pollack, and Wenger. Here we show a colorful genealization of their theorem which confirms a conjecture of Arocha, Bracho, and Montejano. The proof is topological and uses the Borsuk-Ulam theorem.