A Remark on the Nullness of Boundaries of Uniformly Prox-Regular Sets in Banach Spaces

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.