Random polytopes in convex bodies: Bridging the gap between extremal containers

TL;DR

This paper studies the asymptotic behavior of random polytopes formed from points sampled in block-beta distributions, revealing how their properties interpolate between classical smooth and polytopal convex bodies.

Contribution

It introduces a new family of random polytopes with interpolative properties, providing explicit growth rates for facets and faces, bridging classical extremal cases.

Findings

Expected number of facets grows at explicit rates depending on model parameters

Expected number of faces of arbitrary dimensions analyzed for uniform distribution

Volume difference between polytopes and convex bodies characterized

Abstract

We investigate the asymptotic properties of random polytopes arising as convex hulls of $n$ independent random points sampled from a family of block-beta distributions. Notably, this family includes the uniform distribution on a product of Euclidean balls of varying dimensions as a key example. As $n\to\infty$, we establish explicit growth rates for the expected number of facets, which depend in a subtle way on the the underlying model parameters. For the case of the uniform distribution, we further examine the expected number of faces of arbitrary dimensions as well as the volume difference. Our findings reveal that the family of random polytopes we introduce exhibits novel interpolative properties, bridging the gap between the classical extremal cases observed in the behavior of random polytopes within smooth versus polytopal convex containers.