An elementary exposition to topological overlap in the plane

TL;DR

This paper provides an accessible, self-contained explanation of Gromov's topological overlap theorem in the plane, including a weighted generalization using von Neumann's minimax theorem.

Contribution

It offers an elementary exposition of Gromov's argument and introduces a weighted version leveraging probability distributions and duality principles.

Findings

Elementary proof of Gromov's topological overlap in the plane

Weighted generalization using probability distributions

Application of von Neumann's minimax theorem

Abstract

The aim of this text is to provide an elementary and self-contained exposition of Gromov's argument on topological overlap (the presentation is based on Gromov's work, as well as two follow-up papers of Matousek and Wagner, and of Dotterrer, Kaufman and Wagner). We also discuss a simple generalization in which the vertices are weighted according to some probability distribution. This allows to use von Neumann's minimax theorem to deduce a dual statement.