Anisotropic curvature measures and uniqueness of convex bodies

TL;DR

This paper proves that convex bodies with constant anisotropic curvature measures are necessarily rescaled and translated Wulff shapes, generalizing previous theorems and resolving a longstanding conjecture in convex geometry.

Contribution

It establishes a new characterization of convex bodies based on anisotropic curvature measures, extending classical results to a broader setting.

Findings

Convex bodies with constant anisotropic curvature measures are Wulff shapes.

The result generalizes Schneider's theorem from 1979.

It resolves a conjecture by Andrews and Wei from 2017.

Abstract

We prove that an arbitrary convex body $C \subseteq \mathbf{R}^{n+1} $, whose $ k $-th anisotropic curvature measure (for $ k =0, \ldots , 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).