Intersection theorems for closed convex sets and applications (Q257188)

By using a standard separation theorem, the author proves first the convex KKM theorem (the Ky Fan KKM principle, when the involved mapping is convex-valued). Then, this result provides simple proofs of several fundamental results from the KKM theory: fixed point or coincidence theorems, variational inequalities, minimax inequalities, etc. The used method avoids deeper arguments of topological essence such as the Brouwer fixed point theorem or the Sperner's lemma and underlines the crucial role played by convexity.