FIXED-POINT THEOREMS FOR A CONTROLLED WITHDRAWAL OF THE CONVEXITY OF THE VALUES OF A SET-VALUED MAP

The question of the extent of the possible weakening of the convexity condition for the values of set-valued maps in the classical fixed-poi nt theorems of Kakutani, Bohnenblust-Karlin, and Gliksberg is discusse d. For: an answer, one associates with each closed subset P of a Banac h space a numerical function alpha(P) : (0, infinity) --> [0, infinity ), which is called the function of non-convexity of P. The closer alph a(P) is to zero, the 'more convex' is P. The equality alpha(P) = 0 is equivalent to the convexity of P. Results on selections, approximation s, and fixed points for set-valued maps F of finite- and infinite-dime nsional paracompact sets are established in which the equality alpha(F )(x) = 0 is replaced by conditions of the kind: ''alpha(F)(x) is less than 1''. Several formalizations of the last condition are compared an d the topological stability of constraints of this type is shown.