Intersection theorems for closed convex sets and applications

zbMATH Open1341.52002arXiv1501.05813MaRDI QIDQ257188

Publication date: 15 March 2016

Published in: Missouri Journal of Mathematical Sciences (Search for Journal in Brave)

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.

Full work available at URL: https://arxiv.org/abs/1501.05813

zbMATH Keywords

variational inequalities Hahn-Banach theorem coincidence minimization of functionals intersection theorems convex KKM theorem fixed points for von Neumann relations Markov-Kakutani fixed point theorem separation of convex sets systems of nonlinear inequalities

Mathematics Subject Classification ID

Fixed-point theorems (47H10) Set-valued operators (47H04) Applications of operator theory in optimization, convex analysis, mathematical programming, economics (47N10) Convex sets in topological vector spaces (aspects of convex geometry) (52A07) Topological consequences of geometric convexity (32F27) Analytical consequences of geometric convexity (vanishing theorems, etc.) (32F32)