From Simplices to Abstract Convex Spaces: A brief history of the KKM theory

Abstract

We review briefly the history of the KKM theory from the original KKM theorem on simplices in 1929 to the birth of the new partial KKM spaces by the following steps.
(1) We recall some early equivalent formulations of the Brouwer fixed point theorem and the KKM theorem.
(2) We summarize Fan’s foundational works on the KKM theory from 1960s to 1980s.
(3) We note that, in 1983-2005, basic results in the theory were extended to convex spaces by Lassonde, to H-spaces by Horvath, and to G-convex spaces due to Park.
(4) In 2006, we introduced the concept of abstract convex spaces (E, D; Γ) on which we can construct the KKM theory. Moreover, abstract convex spaces satisfying an abstract form of the KKM theorem were called partial KKM spaces. Now the KKM theory becomes the study of such spaces.
(5) Various properties hold for partial KKM spaces and many new types of such spaces are introduced. We state a metatheorem for common properties or applications of such spaces.
(6) Finally, we introduce the partial KKM space versions of the von Neumann minimax theorem, the von Neumann intersection lemma, the Nash equilibrium theorem, and the Himmelberg fixed point theorem.