Vectorial analogues of Cauchy's surface area formula

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.