Angle sums of random polytopes

TL;DR

This paper explicitly computes expected sums of conic intrinsic volumes and angles for two families of random polytopes, revealing new formulas and invariance properties, with implications for Gaussian projections and affine transformations.

Contribution

It provides explicit formulas for expected angle sums of Gaussian polytopes and convex hulls of random walks, and establishes invariance under affine transformations for these sums.

Findings

Expected sums expressed via angles of regular simplices and Stirling numbers.

Invariance of Grassmann angle sums under affine transformations.

Angle sums computed for Gaussian projections of polyhedral sets.

Abstract

For two families of random polytopes we compute explicitly the expected sums of the conic intrinsic volumes and the Grassmann angles at all faces of any given dimension of the polytope under consideration. As special cases, we compute the expected sums of internal and external angles at all faces of any fixed dimension. The first family are the Gaussian polytopes defined as convex hulls of i.i.d. samples from a non-degenerate Gaussian distribution in $\mathbb R^d$. The second family are convex hulls of random walks with exchangeable increments satisfying certain mild general position assumption. The expected sums are expressed in terms of the angles of the regular simplices and the Stirling numbers, respectively. There are non-trivial analogies between these two settings. Further, we compute the angle sums for Gaussian projections of arbitrary polyhedral sets, of which the Gaussian polytopes are a special case. Also, we show that the expected Grassmann angle sums of a random polytope with a rotationally invariant law are invariant under affine transformations. Of independent interest may be also results on the faces of linear images of polyhedral sets. These results are well known but it seems that no detailed proofs can be found in the existing literature.