Vectorial analogues of Cauchy’s surface area formula

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.