Abstract
To each closed subset P of a Banach space, a real function ⊐ characterizing the nonconvexity of this set is associated. Inequalities of the type α P(.),)< \nomathbreak 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 ∈-neighborhoods of sets.
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Repovš, D., Semenov, P.V. On the Relation between the Nonconvexity of a Set and the Nonconvexity of Its ɛ-Neighborhoods. Mathematical Notes 70, 221–232 (2001). https://doi.org/10.1023/A:1010258909731
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DOI: https://doi.org/10.1023/A:1010258909731