Some of Sion's heirs and relatives

摘要

If one adds one extra assumption to the classical Knaster- Kuratowski-Mazurkiewicz (KKM) theorem, namely that the sets F (i) are convex, one gets the "Elementary" KKM theorem; the name is due to A. Granas and M. Lassonde (1995) who gave a simple proof of the Elementary KKM theorem and showed that despite being "elementary," it is powerful and versatile. It is shown here that this Elementary KKM theorem is equivalent to Klee's theorem, the Elementary Alexandroff- Pasynkov theorem, the Elementary Ky Fan theorem and the Sion-von Neumann minimax theorem, as well as a few other classical results with an extra convexity assumption; hence the adjective "elementary." The Sion-von Neumann minimax theorem itself can be proved by simple topological arguments using connectedness instead of convexity. This work answers a question of Professor Granas regarding the logical relationship between the Elementary KKM theorem and the Sion-von Neumann minimax theorem.