On a fixed point theorem of Ky Fan

We generalize a theorem of Ky Fan about the nearest distance between a closed convex set D in a Banach space E and its image by a function f : D --> E in several directions : (1) for noncompact sets D, when f (D) precompact; (2) for compact D and upper semicontinuous multifunction f; and more generally, (3) for noncompact D and upper semicontinuous multifunction f with f(D) Hausdorff precompact. In particular, we prove a version of the fixed point theorem of Kakutani-Ky Fan for multifunctions, whose values are convex closed bounded, thus not necessarily compact.