On the geometry of random convex sets between polytopes and zonotopes

TL;DR

This paper investigates a class of random convex sets that interpolate between polytopes and zonotopes, analyzing their geometric properties using asymptotic formulas for support functions and mean width.

Contribution

It introduces a new model for random convex sets based on moments of order statistics of random vectors, generalizing existing models for random polytopes.

Findings

Asymptotic formulas for support function and mean width are derived.

The model encompasses and extends standard random polytope models.

Results are sharp up to absolute constants.

Abstract

In this work we study a class of random convex sets that "interpolate" between polytopes and zonotopes. These sets arise from considering a $q^{th}$-moment ($q\geq 1$) of an average of order statistics of $1$-dimensional marginals of a sequence of $N\geq n$ independent random vectors in $\mathbb R^n$. We consider the random model of isotropic log-concave distributions as well as the uniform distribution on an $\ell_p^n$-sphere ($1\leq p < \infty$) with respect to the cone probability measure, and study the geometry of these sets in terms of the support function and mean width. We provide asymptotic formulas for the expectation of these geometric functionals which are sharp up to absolute constants. Our model includes and generalizes the standard one for random polytopes.