The mean width of circumscribed random polytopes

TL;DR

This paper investigates how the mean width of random polytopes circumscribed around a convex body in high-dimensional space converges to that of the body, providing bounds and asymptotic formulas.

Contribution

It establishes optimal bounds and precise asymptotics for the mean width difference between a convex body and its circumscribed random polytopes.

Findings

Optimal order bounds for mean width difference

Asymptotic formulas for simplicial polytopes

Convergence rates as number of halfspaces increases

Abstract

For a given convex body K in $R^d$, a random polytope $K^{(n)}$ is defined (essentially) as the intersection of $n$ independent closed halfspaces containing $K$ and having an isotropic and (in a specified sense) uniform distribution. We prove upper and lower bounds, of optimal orders, for the difference of the mean widths of $K^{(n)}$ and K, as n tends to infinity. For a simplicial polytope P, a precise asymptotic formula for the difference of the mean widths of $P^{(n)}$ and P is obtained.