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

Abstract

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