Anisotropic curvature measures and uniqueness of convex bodies

Abstract

We prove that an arbitrary convex body $C\subseteq {\mathbf{R}}^{n+1}$, whose $k$-th anisotropic curvature measure (for $k=0,\dots ,n-1$) is a multiple constant of the anisotropic perimeter of C, must be a rescaled and translated Wulff shape.This result provides a generalization of a theorem of Schneider (1979) and resolves a conjecture of Andrews and Wei (2017).

arXiv commentThis preprint is not going to be published, as a solution of the conjecture mentioned in the abstract is contained in Theorem 2.43 of "Daniel Hug. Measures, curvatures and currents in convex geometry. Habilitationsschrift, Albert-Ludwigs-Universitaet, Freiburg im Breisgau, December 1999. https://doi.org/10.6094/UNIFR/167349". This submission is superseded by arXiv:2204.04909

References in corpus (1)

• Uniqueness of singular convex hypersurfaces with lower bounded k-th mean curvature