  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.
