Geodesic
Continuity and Selections of the Intersection Operator Applied to Nonconvex Sets
Journal of convex analysis, Tome 22 (2015) no. 4, pp. 939-962
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\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.
Classification : 41A50, 41A65, 52A21
Mots-clés : Support condition, weak convexity, proximal regularity, quasiball, multifunction, selection 
@article{JCA_2015_22_4_JCA_2015_22_4_a2,
     author = {G. E. Ivanov},
     title = {Continuity and {Selections} of the {Intersection} {Operator} {Applied} to {Nonconvex} {Sets}},
     journal = {Journal of convex analysis},
     pages = {939--962},
     year = {2015},
     volume = {22},
     number = {4},
     url = {http://geodesic.mathdoc.fr/item/JCA_2015_22_4_JCA_2015_22_4_a2/}
}
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G. E. Ivanov. Continuity and Selections of the Intersection Operator Applied to Nonconvex Sets. Journal of convex analysis, Tome 22 (2015) no. 4, pp. 939-962. http://geodesic.mathdoc.fr/item/JCA_2015_22_4_JCA_2015_22_4_a2/