true for any family { φ$_{1}$, . . . , φ$_{k}$ } ⊂ F with k = n − 1 and let { φ$_{1}$, . . . , φ$_{n}$ } ⊂ F be arbitrary. For any x ∈ ⋂ n − 1 i =1 F ix ( φ$_{i}$ ) , φ$_{n}$ ( x ) = φ$_{n}$ ( φ$_{i}$ ( x )) = φ$_{i}$ ( φ$_{n}$ ( x )) for all i = 1 , . . . , n − 1 , i.e., φ$_{n}$ ( x ) ∈ ⋂ n − 1 i =1 F ix ( φ$_{i}$ ) . Thus φ$_{n}$ maps the non-empty compact convex set ⋂ n − 1 i =1 F ix ( φ$_{i}$ ) into itself. By Corollary 21 again, it has a fixed point ¯ x = φ$_{n}$ (¯ x ) ∩ ⋂ n − 1 i =1 F ix ( φ$_{i}$ ) , i.e., ¯ x ∈ ⋂ n i $_{=1}$F ix ( φ$_{i}$ ) . It is well established that the Markov-Kakutani fixed point theorem implies the Hahn-Banach theorem (Kakutani [16]). The two results are indeed equivalent (for a short and elegant proof of the converse, see D. Werner [21]). Since we derived here the convex KKM theorem from the theorem on the separation of convex sets, we have thus established the equivalence of the Hahn-Banach theorem, Klee’s intersection theorem, the convex KKM theorem, the fixed point theorem for von Neumann relations, the Sion-von Neumann minimax theorem, and the Markov-Kakutani fixed point theorem. Although the convex KKM theorem is a particular instance of the KKM principle of Ky Fan, and since the fixed point and coincidence properties in Theorems 9 and 12 are special cases of similar results for so-called F and F ∗ maps (see [2, 3, 4]), the interest here resides in the use of simple arguments of convexity rather than the Brouwer fixed point theorem or Sperner’s lemma. It would be most interesting to know if the Ky Fan KKM principle (or any of its equivalent results) can be derived directly from the convex KKM theorem. In other words, can any of the question marks below be settled? (The smaller arrows are established; FPT stands for Fixed Point Theorem.) 20 5 Concluding Remarks