Abstract

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

Author information

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Grigorii E. Ivanov

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

Keywords

  • Support condition
  • weak convexity
  • proximal regularity
  • quasiball
  • multifunction
  • selection

Mathematics Subject Classification

41A50, 41A65, 52A21

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Suggested citation

G. E. Ivanov. “Continuity and Selections of the Intersection Operator Applied to Nonconvex Sets.” Journal of Convex Analysis 22 (2015), No. 4, 939–962. https://doi.org/10.68381/jca22050

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Published by Heldermann Verlag, 2015. Rights now held by Banach Press LLC.