The endpoint multilinear Kakeya theorem via the Borsuk--Ulam theorem

TL;DR

This paper provides a simplified, self-contained proof of Guth's endpoint multilinear Kakeya theorem by employing the Borsuk-Ulam theorem, avoiding complex algebraic topology techniques.

Contribution

The authors present an alternative proof of Guth's theorem that simplifies the approach by replacing algebraic topology with the Borsuk-Ulam theorem.

Findings

Proof avoids sophisticated algebraic topology

Utilizes Borsuk-Ulam theorem for the proof

Simplifies understanding of the endpoint multilinear Kakeya theorem

Abstract

We give an essentially self-contained proof of Guth's recent endpoint multilinear Kakeya theorem which avoids the use of somewhat sophisticated algebraic topology, and which instead appeals to the Borsuk-Ulam theorem.