On the Relation between the Nonconvexity of a Set and the Nonconvexity of Its $\varepsilon$-Neighborhoods

To each closed subset $P$ of a Banach space, a real function $\alpha _P$ characterizing the nonconvexity of this set is associated. Inequalities of the type $\alpha _P(\cdot )<1$ ensure good topological properties of the set $P$, such as contractibility, the property of being an extensor, etc. In this paper, examples of sets whose nonconvexity functions substantially differ from the nonconvexity functions of arbitrarily small neighborhoods of these sets are constructed. On the other hand, it is shown that, in uniformly convex Banach spaces, conditions of the type “the function of nonconvexity is less than one” are stable with respect to taking $\varepsilon$-neighborhoods of sets.