Published Paper

Inserted: 22 aug 2023

Journal: Advances in Mathematics
Year: 2022
Doi: https://doi.org/10.1016/j.aim.2022.108802

Abstract:

Extending the celebrated results of Alexandrov (1958) and Korevaar--Ros (1988) for smooth sets, as well as the results of Schneider (1979) and the first author (1999) for arbitrary convex bodies, we obtain for the first time the characterization of the isoperimetric sets of a uniformly convex smooth finite-dimensional normed space (i.e.\ Wulff shapes) in the non-smooth and non-convex setting, based on a natural geometric condition involving the curvature measures. More specifically we show, under a weak mean convexity assumption, that finite unions of disjoint Wulff shapes are the only sets of positive reach $ A \subseteq \mathbf{R}^{n+1} $ with finite and positive volume such that, for some $ k \in \{0, \ldots , n-1\}$, the $ k $-th generalized curvature measure $ \Theta^\phi_{k}(A, \cdot) $, which is defined on the unit normal bundle of $ A $ with respect to the relative geometry induced by $ \phi $, is proportional to the top order generalized curvature measure $ \Theta^\phi_{n}(A, \cdot)$. If $ k = n-1 $ the conclusion holds for all sets of positive reach with finite and positive volume. We also prove a related sharp result about the removability of the singularities. This result is based on the extension of the notion of a normal boundary point, originally introduced by Busemann and Feller (1936) for arbitrary convex bodies, to sets of positive reach.

These findings are new even in the Euclidean space.

Several auxiliary and related results are proved, which are of independent interest. They include the extension of the classical Steiner--Weyl tube formula to arbitrary closed sets in a finite dimensional uniformly convex normed vector space, a general formula for the derivative of the localized volume function, which extends and complements recent results of Chambolle--Lussardi--Villa (2021), and general versions of the Heintze--Karcher inequality.