Asymptotic behavior of a set-statistic

The existence theorem of Minkowski for a polytope with given facet normals and areas is adapted to a data-analytic context. More precisely, we show that a centered, random point sample arising from an absolutely continuous distribution in R^d can be uniquely mapped into such a polytope almost surely. With increasing sample size, the sequence of (scaled) polytopes converges almost surely to a limiting convex body that is associated with the underlying distribution. An accompanying central limit theorem is proved using methods from the theory of empirical processes.