The volume of random polytopes circumscribed around a convex body

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.

References in corpus (2)

• The volume of random polytopes circumscribed around a convex body
• Hölder continuity of normal cycles and of support measures of convex bodies

Cited by in corpus (4)

• Variance estimates for random disc-polygons in smooth convex discs
• The volume of random polytopes circumscribed around a convex body
• Interaction of Poisson hyperplane processes and convex bodies
• A Concentration Inequality for Random Polytopes, Dirichlet-Voronoi Tiling Numbers and the Geometric Balls and Bins Problem