Abstract

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.

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Keywords

  • Proximally smooth set
  • uniformly prox-regular set
  • a-convex set
  • Gamma-null set
  • uniformly smooth Banach space
  • uniformly convex Banach space

Mathematics Subject Classification

52A30; 46B99

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Suggested citation

L. Zajíček. “A Remark on the Nullness of Boundaries of Uniformly Prox-Regular Sets in Banach Spaces.” Journal of Convex Analysis 32 (2025), No. 3, 921–926. https://doi.org/10.68381/jca32047

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Published by Heldermann Verlag, 2025. Rights now held by Banach Press LLC.