Convex KKM maps, monotone operators and Minty variational inequalities

TL;DR

This paper explores the relationship between convex KKM maps, monotone operators, and Minty variational inequalities, providing new characterizations and a converse to Minty's theorem in convex analysis.

Contribution

It introduces a novel characterization of monotone operators using convex KKM maps and establishes a converse to Minty's theorem based on solution existence.

Findings

Characterization of monotone operators via convex KKM maps

Equivalence between KKM condition and finite intersection property for convex sets

A new converse to Minty's theorem relating solutions to variational inequalities

Abstract

It is known that for convex sets, the KKM condition is equivalent to the finite intersection property. We use this equivalence to obtain a characterisation of monotone operators in terms of convex KKM maps and in terms of the existence of solutions to Minty variational inequalities. The latter result provides a converse to the seminal theorem of Minty.