Intersection Theorems for Closed Convex Sets and Applications Hichem Ben-El-Mechaiekh Department of Mathematics, Brock University, Saint Catharines, Ontario, Canada, L2S 3A1 To Appear in Missouri J. Math. Sc. (Sumitted: November 9, 2011; Accepted: May 22, 2012) Abstract A number of landmark existence theorems of nonlinear functional anal- ysis 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 topo- logical vector spaces - and a coincidence property for so-called von Neu- mann 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 mini- max principle. Keywords and phrases: Separation of convex sets, intersection theo- rems, convex KKM theorem, fixed points for von Neumann relations, co- incidence, systems of nonlinear inequalities, variational inequalities, min- 1 arXiv:1501.05813v1 [math.FA] 23 Jan 2015