Journal of Convex Analysis 31 (2024), No. 1, 039--050
Copyright Heldermann Verlag 2024
The Extended Exterior Sphere Condition
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
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.
Keywords: Prox-regularity, exterior sphere condition, union of closed balls property, proximal analysis, nonsmooth analysis.
MSC: 49J52, 52A20, 93B27.
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