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-point theorems of Kakutani, Bohnenblust-Karlin, and Gliksberg is discussed. For: an answer, one associates with each closed subset P of a Banach space a numerical function alpha(P) : (0, infinity) --> [0, infinity), which is called the function of non-convexity of P. The closer alpha(P) is to zero, the 'more convex' is P. The equality alpha(P) = 0 is equivalent to the convexity of P. Results on selections, approximations, and fixed points for set-valued maps F of finite- and infinite-dimensional 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 and the topological stability of constraints of this type is shown.