Geodesic
Properties of Sets Admitting Stable $\varepsilon$-Selections
Matematičeskie zametki, Tome 89 (2011) no. 4, pp. 608-613

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Sets in which some convex subsets admit local (global) continuous $\varepsilon$-selections are studied. In particular, it is shown that if, for any number $\varepsilon>0$, 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) $\varepsilon$-selection to an approximatively compact set $M\subset X$, then $x$ is a $\delta$-sun point; if, in addition, $X\in (R)$, then the set of all points nearest to $x$ in $M$ is a singleton.
Keywords: Banach space, acyclic upper semicontinuous $\varepsilon$-selection, approximatively compact set, $\delta$-sun, convex set. 
@article{MZM_2011_89_4_a12,
     author = {I. G. Tsar'kov},
     title = {Properties of {Sets} {Admitting} {Stable} $\varepsilon${-Selections}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {608--613},
     publisher = {mathdoc},
     volume = {89},
     number = {4},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2011_89_4_a12/}
}
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I. G. Tsar'kov. Properties of Sets Admitting Stable $\varepsilon$-Selections. Matematičeskie zametki, Tome 89 (2011) no. 4, pp. 608-613. http://geodesic.mathdoc.fr/item/MZM_2011_89_4_a12/