Vectorial analogues of Cauchy's surface area formula

Daniel Hug; Rolf Schneider

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arXiv · 2307.12257

Vectorial analogues of Cauchy's surface area formula

Daniel Hug·Rolf Schneider

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Abstract

Cauchy's surface area formula says that for a convex body $K$ in $n$-dimensional Euclidean space the mean value of the $(n-1)$-dimensional volumes of the orthogonal projections of $K$ to hyperplanes is a constant multiple of the surface area of $K$. We prove an analogous formula, with the volumes of the projections replaced by their moment vectors. This requires to introduce a new vector-valued valuation on convex bodies.

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Daniel Hug, Rolf Schneider. 2023-07-23. Vectorial analogues of Cauchy's surface area formula. https://arxiv.org/abs/2307.12257

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