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Journal of Convex Analysis 22 (2015), No. 4, 939--962
Copyright Heldermann Verlag 2015

Continuity and Selections of the Intersection Operator Applied to Nonconvex Sets

Grigorii E. Ivanov
Dept. of Higher Mathematics, Moscow Institute of Physics and Technology, Institutski str. 9, Dolgoprudny 141700, Russia


\def\S{{\mathcal{S}}} For a convex body $C$ in a Banach space $E$ we consider the class $\S(C)$ of closed sets $A\subset E$ satisfying the support condition with respect to $C$. If $C$ is a ball with radius $r$, then $\S(C)$ is exactly the class of uniformly $r$-prox-regular sets. We prove that the intersection operator $(A,C)\mapsto A\cap C$ is uniformly Hausdorff continuous and has a uniformly continuous selection on the family of pairs $(A,C)$ such that $C$ is closed and uniformly convex, $rA\in\S(C)$ with $r\in(0,1)$, and $A\cap C\ne\emptyset$. We also deduce some new sufficient condition for affirmative solution of the splitting problem for selections.

Keywords: Support condition, weak convexity, proximal regularity, quasiball, multifunction, selection.

MSC: 41A50, 41A65, 52A21

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