Properties of sets admitting stable ɛ-selections
Игорь Германович Царьков · Mathematical Notes · 2011
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.