Anisotropic area measures of convex bodies

TL;DR

This paper introduces anisotropic area and support measures for convex bodies, generalizing classical concepts by incorporating a gauge body, and characterizes bodies with proportional measures as k-tangential bodies.

Contribution

It develops a new framework of anisotropic measures based on gauge bodies and establishes a characterization theorem for bodies with proportional measures.

Findings

Introduction of anisotropic area and support measures.

Characterization of bodies with proportional measures as k-tangential bodies.

Extension of relative differential geometry concepts to convex bodies.

Abstract

Motivated by the relative differential geometry, where the Euclidean normal vector of hypersurfaces is generalized by a relative normalization, we introduce anisotropic area measures of convex bodies, constructed with respect to a gauge body. Together with the anisotropic curvature measures, they are special cases of the newly introduced anisotropic support measures. We show that a convex body in ${\mathbb R}^n$, for which the anisotropic area measure of some order $k\in\{0,\dots,n-2\}$ is proportional to the area measure of order $n-1$, must be a $k$-tangential body of the gauge body.