Intersection Theorems for Closed Convex Sets and Applications

TL;DR

This paper presents a unified approach to fundamental theorems in nonlinear functional analysis using convex set separation and KKM principles, avoiding complex topological theorems.

Contribution

It extends the convex KKM principle to arbitrary topological vector spaces and establishes its equivalence with key theorems like Hahn-Banach and fixed point theorems.

Findings

Unified proof of existence theorems via convex separation

Extension of KKM principle to general topological vector spaces

Equivalence of convex KKM principle with major functional analysis theorems

Abstract

A number of landmark existence theorems of nonlinear functional analysis follow in a simple and direct way from the basic separation of convex closed sets in finite dimension via elementary versions of the Knaster-Kuratowski-Mazurkiewicz principle - which we extend to arbitrary topological vector spaces - and a coincidence property for so-called von Neumann relations. The method avoids the use of deeper results of topological essence such as the Brouwer fixed point theorem or the Sperner's lemma and underlines the crucial role played by convexity. It turns out that the convex KKM principle is equivalent to the Hahn-Banach theorem, the Markov-Kakutani fixed point theorem, and the Sion-von Neumann minimax principle.