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 $\alpha_P\colon(0,\infty)\to[0,\infty)$, which is called the function of non-convexity of $P$. The closer $\alpha_P$ is to zero, the 'more convex' is $P$. The equality $\alpha_P\equiv 0$ 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 $\alpha_{F(x)}\equiv 0$ 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.