Some elementary general principles of convex analysis

In a recent paper [6], the authors presented a new geometric approach in the theory of minimax inequalities, which has numerous applications in different areas of mathematics. In this note, we complement and elucidate the above approach within the context of complete metric spaces. More precisely, we concentrate on super-reflexive Banach spaces and show that a large part of the theory of these spaces (and, in particular, Hilbert spaces) can be obtained in a very elementary way, without using weak topology or compactness. In Section 2, we give an elementary proof of the basic intersection property of closed convex bounded sets and give applications in Section 3. In Section 4, we describe the KKM property and in Section 5 we prove the fundamental intersection property of KKM-maps with closed convex bounded values. The remaining sections are devoted to applications to variational inequalities, theory of games, systems of inequalities and maximal monotone operator theory, respectively.