Abstract

We prove that the complement of a closed set S satisfying an extended exterior sphere condition is nothing but the union of closed balls with common radius. This generalizes Theorem 3 of F. Nacry and L. Thibault [Distance function associated to a prox-regular set, Set-Valued Var. Analysis 30 (2022) 731–750] where the set S is assumed to be prox-regular, a property stronger than the extended exterior sphere condition. We also provide a sufficient condition for the equivalence between prox-regularity and the extended exterior sphere condition that generalizes previous work of C. Nour, R. J. Stern and J. Takche [Proximal smoothness and the exterior sphere condition, J. Convex Analysis 16/2 (2009) 501–514] to the case in which S is not necessarily regular closed.

Author information

Contact details are reproduced from the original publication and may be historical.

Chadi Nour

Dept. of Computer Science and Mathematics, Lebanese American University, Byblos, Lebanon

cnour@lau.edu.lb

Jean Takche

Dept. of Computer Science and Mathematics, Lebanese American University, Byblos, Lebanon

jtakchi@lau.edu.lb

Keywords

  • Prox-regularity
  • exterior sphere condition
  • union of closed balls property
  • proximal analysis
  • nonsmooth analysis

Mathematics Subject Classification

49J52, 52A20, 93B27

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Suggested citation

C. Nour, J. Takche. “The Extended Exterior Sphere Condition.” Journal of Convex Analysis 31 (2024), No. 1, 39–50. https://doi.org/10.68381/jca31003

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Published by Heldermann Verlag, 2024. Rights now held by Banach Press LLC.