Sums of sections, surface area measures, and the general Minkowski problem

Abstract

For $2 \leq k \leq d - 1$, the $k$-th mean section body, $M_k(K)$, of a convex body $K$ in $\mathbb{R}^d$, is the Minkowski sum of all its sections by $k$-dimensional flats. We will show that the characterization of these mean section bodies is equivalent to the solution of the general Minkowski problem, namely that of giving the characteristic properties of those measures on the unit sphere which arise as surface area measures (of arbitrary degree) of convex bodies. This equivalence arises from an analysis of Berg's solution of the Christoffel problem. We will see how the functions introduced by Berg yield an integral representation of the support function of $M_k(K)$ in terms of the $(d + 1 - k)$-th surface area measure of $K$. Our results will be obtained using Fourier transform techniques which also yield a stability version of the fact that $M_k(K)$ determines $K$ uniquely.