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

Abstract:

Let $K$ be a convex body in $\mathbb {R}^d$, let $j\in \{1, \ldots , d-1\}$, 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 $\mathbb {R}^d$ 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 $K$.