Intrinsic volumes of random polytopes with vertices on the boundary of a convex body

TL;DR

This paper extends the asymptotic analysis of intrinsic volumes of random polytopes formed from boundary points of a convex body, relaxing smoothness conditions to include bodies where a ball rolls freely.

Contribution

It generalizes previous results by deriving asymptotic formulas for intrinsic volumes without requiring the boundary to be a smooth manifold.

Findings

Asymptotic formulas for intrinsic volumes of random polytopes

Extension to convex bodies with rolling ball condition

Broader applicability to non-smooth convex bodies

Abstract

Let $K$ be a convex body in $\R^d$, let $j\in\{1, ..., d-1\}$, and let $\varrho$ be a positive and continuous probability density function with respect to the $(d-1)$-dimensional Hausdorff measure on the boundary $\partial K$ of $K$. Denote by $K_n$ the convex hull of $n$ points chosen randomly and independently from $\partial K$ according to the probability distribution determined by $\varrho$. For the case when $\partial K$ is a $C^2$ submanifold of $\R^d$ with everywhere positive Gauss curvature, M. Reitzner proved an asymptotic formula for the expectation of the difference of the $j$th intrinsic volumes of $K$ and $K_n$, as $n\to\infty$. In this article, we extend this result to the case when the only condition on $K$ is that a ball rolls freely in $K$.