2022/01/01 by Daniel Hug, Hug, Daniel, Mario Santilli +1 · 3 citations
Mathematics · #26B25 #28A75 #49Q10 #52A30 #53C45 #53C60 #53C65 #Analysis of PDEs (math.AP) #Analytic and geometric function theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Metric Geometry (math.MG) #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2204.04909
openalex publication_date 2022/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 the natural geometric condition involving the curvature measures. More specifically we show, under a natural mean-convexity assumption, that finite unions of disjoint Wulff shapes are the only sets of positive reach A⊆Rn+1 with finite and positive volume such that, for some k∈ \0,…,n−1\, the k-th generalized curvature measure Θϕn(A,⋅), which is defined on the unit normal bundle of A with respect to the relative geometry induced by ϕ, is proportional to Θϕn(A,⋅). 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 results are new even in the Euclidean space. Several auxiliary and related results are also 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 and a general formula for the derivative of the localized volume function, which extends and complements recent results of Chambolle--Lussardi--Villa (2021).