On a transversal theorem of Montejano and Karasev

TL;DR

This paper presents a simplified proof of a theorem on $k$-dimensional transversals to convex sets, using the Borsuk-Ulam theorem, and extends the original result in some cases.

Contribution

A new, simpler proof of Montejano and Karasev's theorem employing the Borsuk-Ulam theorem, with potential improvements in specific cases.

Findings

Simplified proof of the original theorem.

Potentially stronger results in certain cases.

Application of the Borsuk-Ulam theorem to convex set transversals.

Abstract

We give a new proof of a theorem of Montejano and Karasev regarding $k$-dimensional transversals to small families of convex sets. While their proof uses technical algebraic and topological tools, our proof is a simple application of the Borsuk-Ulam theorem. Additionally, in certain cases our result is stronger than the Montejano-Karasev theorem.