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arXiv · 2506.08803

Anisotropic area measures of convex bodies

Rolf Schneider
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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.

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Rolf Schneider. 2025-06-10. Anisotropic area measures of convex bodies. https://arxiv.org/abs/2506.08803

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