Properties of Sets Admitting Stable $\varepsilon$-Selections

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.