Abstract
Sets in which some convex subsets admit local (global) continuous ɛ-selections are studied. In particular, it is shown that if, for any number ɛ > 0, some neighborhood O(x) of a point x in a Banach space X contains a dense (in O(x)) convex set K admitting an upper semicontinuous acyclic (in particular, continuous single-valued) ɛ-selection to an approximatively compact set M ⊂ X, then x is a δ-sun point; if, in addition, X ∈ (R), then the set of all points nearest to x in M is a singleton.
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Original Russian Text © I. G. Tsar’kov, 2011, published in Matematicheskie Zametki, 2011, Vol. 89, No. 4, pp. 608–613.
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Tsar’kov, I.G. Properties of sets admitting stable ɛ-selections. Math Notes 89, 572–576 (2011). https://doi.org/10.1134/S0001434611030291
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DOI: https://doi.org/10.1134/S0001434611030291