Using the KKM theorem

TL;DR

This paper surveys various generalizations of the KKM theorem, highlighting their applications in geometry, combinatorics, and fair division, and introduces new results and open problems in the field.

Contribution

It provides a comprehensive overview of KKM theorem extensions, includes new results, and discusses open problems in related areas.

Findings

Multiple generalizations of the KKM theorem are applicable to diverse mathematical problems.

New KKM-type results are presented, expanding the theorem's utility.

Open problems in the field are identified for future research.

Abstract

The KKM theorem, due to Knaster, Kuratowski, and Mazurkiewicz in 1929, is a fundamental result in fixed-point theory, which has seen numerous extensions and applications. In this paper we survey old and recent generalizations of the KKM theorem and their applications in the areas of piercing numbers, mass partition, fair division, and matching theory. We also give a few new results utilizing KKM-type theorems, and discuss related open problems.