Journal of Convex Analysis 32 (2025), No. 3, 921--926
Copyright Heldermann Verlag 2025
A Remark on the Nullness of Boundaries of Uniformly Prox-Regular Sets in Banach Spaces
Ludek Zajícek
Charles University, Faculty of Mathematics and Physics, Praha, Czech Republic
zajicek@karlin.mff.cuni.cz
We prove that, in any separable uniformly smooth Banach space, the boundary of every a-convex set (in the Efimov-Stechkin sense) is Γ-null (in the Lindenstrauss-Preiss sense). Using well-known results, we obtain that the same is true for proximally smooth sets (in the sense of Clarke, Stern and Wolenski) and for uniformly prox-regular sets (in the sense of Poliquin, Rockafellar and Thibault) in any separable Banach space which is both uniformly convex and uniformly smooth.
Keywords: Proximally smooth set, uniformly prox-regular set, a-convex set, Gamma-null set, uniformly smooth Banach space, uniformly convex Banach space.
MSC: 52A30; 46B99.
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