A simpler proof of the Boros-F\"uredi-B\'ar\'any-Pach-Gromov theorem

TL;DR

This paper presents a simplified and nearly elementary proof of a theorem concerning the multiplicity of covering points by simplices in multi-dimensional space, making the original complex proof more accessible.

Contribution

It offers a new, simpler proof of the Boros-F"uredi-Bárány-Pach-Gromov theorem, reducing complexity and enhancing understanding of the covering multiplicity in geometry.

Findings

Provided a more accessible proof of the theorem

Simplified the understanding of covering multiplicity in high dimensions

Potentially broadens applicability of the theorem in geometric analysis

Abstract

A short and almost elementary proof of the Boros-F\"uredi-B\'ar\'any-Pach-Gromov theorem on the multiplicity of covering by simplices in $\mathbb R^d$ is given.