Vectorial analogues of Cauchy's surface area formula

TL;DR

This paper introduces a vector-valued analogue of Cauchy's surface area formula, replacing projection volumes with moment vectors, and develops a new valuation on convex bodies.

Contribution

It presents a novel vector-valued version of Cauchy's formula and introduces a new vector-valued valuation on convex bodies.

Findings

Established a vectorial analogue of Cauchy's surface area formula.

Defined a new vector-valued valuation on convex bodies.

Extended classical geometric formulas to vector-valued contexts.

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.