Vectorial analogues of Cauchy's surface area formula

2023/07/23 by Daniel Hug, Hug, Daniel, Rolf Schneider +1
Mathematics · #FOS: Mathematics #Metric Geometry (math.MG) #Point processes and geometric inequalities #Primary 52A20 #Secondary 53C65

paper · pdf · doi:10.48550/arxiv.2307.12257

openalex publication_date 2023/07/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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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