The volume of random polytopes circumscribed around a convex body

TL;DR

This paper investigates the asymptotic behavior of the volume difference and variance of random polytopes circumscribed around a convex body, providing new formulas and bounds for these geometric quantities.

Contribution

It introduces asymptotic formulas for the expected volume difference and upper bounds on the volume variance of random circumscribed polytopes around convex bodies.

Findings

Asymptotic formula for the expectation of volume difference

Upper bound on the variance of the volume

Results derived for convex bodies with a rolling ball property

Abstract

Let $K$ be a convex body in $\mathbb{R}^d$ which slides freely in a ball. Let $K^{(n)}$ denote the intersection of $n$ closed half-spaces containing $K$ whose bounding hyperplanes are independent and identically distributed according to a certain prescribed probability distribution. We prove an asymptotic formula for the expectation of the difference of the volumes of $K^{(n)}$ and $K$, and an asymptotic upper bound on the variance of the volume of $K^{(n)}$. We achieve these results by first proving similar statements for weighted mean width approximations of convex bodies that admit a rolling ball by inscribed random polytopes and then by polarizing these results.