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

Authors : Sehie PARK

Pages : 1-12

View : 20 | Download : 10

Publication Date : 2018-01-19

Article Type : Review Paper

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. insert ignore into journalissuearticles values(1); We recall some early equivalent formulations of the Brouwer fixed point theorem and the KKM theorem. insert ignore into journalissuearticles values(2); We summarize Fan’s foundational works on the KKM theory from 1960s to 1980s. insert ignore into journalissuearticles values(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. insert ignore into journalissuearticles values(4); In 2006, we introduced the concept of abstract convex spaces insert ignore into journalissuearticles values(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. insert ignore into journalissuearticles values(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. insert ignore into journalissuearticles values(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.
Keywords : KM space, partial KKM principle, minimax theorem, equilibrium theorem, fixed point theorem