Random ball-polytopes in smooth convex bodies

TL;DR

This paper investigates how well smooth convex bodies can be approximated by random ball-polytopes, revealing that the expected number of facets converges for certain bodies, unlike classical polytopes.

Contribution

It introduces a probabilistic model for approximating smooth convex bodies with random ball-polytopes and analyzes their asymptotic facet count behavior.

Findings

Expected facet number converges for smooth bodies as n increases.

Sufficiently round bodies behave similarly to classical polytope approximations.

Unique finite limit observed when approximating a unit ball with unit radius ball-polytopes.

Abstract

We study approximations of smooth convex bodies by random ball-polytopes. We examine the following probability model: let $K\subset{\bf R}^d$ be a convex body such that $K$ slides freely in a ball of radius $R>0$ and has $C^2$ smooth boundary. Let $x_1,\ldots, x_n$ be i.i.d. uniform random points in $K$. For $r\geq R$, let $K^r_{(n)}$ denote the intersection of all radius $r$ closed balls that contain $x_1,\ldots, x_n$. Then $K^r_{(n)}$ is a (uniform) random ball-polytope (of radius $r$) in $K$. We study the asymptotic properties of the expectation of the number of facets of $K_{(n)}^r$ as $n\to\infty$. While sufficiently round convex bodies behave in a similar way with respect to random approximation by ball-polytopes as to classical polytopes, an interesting phenomenon can be observed when a unit ball is approximated by unit radius random ball-polytopes: the expected number of facets approaches a finite limit as $n\to\infty$.