Intrinsic volumes of inscribed random polytopes in smooth convex bodies

Abstract

Let $K$ be a $d$ dimensional convex body with a twice continuously differentiable boundary and everywhere positive Gauss-Kronecker curvature. Denote by $K_n$ the convex hull of $n$ points chosen randomly and independently from $K$ according to the uniform distribution. Matching lower and upper bounds are obtained for the orders of magnitude of the variances of the $s$-th intrinsic volumes $V_s(K_n)$ of $K_n$ for $s\in \{1,...,d\}$. Furthermore, strong laws of large numbers are proved for the intrinsic volumes of $K_n$. The essential tools are the Economic Cap Covering Theorem of Bárány and Larman, and the Efron-Stein jackknife inequality.

References in corpus (2)

• Variance asymptotics and central limit theorems for generalized growth processes with applications to convex hulls and maximal points
• Poisson polytopes

Cited by in corpus (5)

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• Intrinsic and Dual Volume Deviations of Convex Bodies and Polytopes
• Variance estimates for random disc-polygons in smooth convex discs
• Random Inscribed Polytopes Have Similar Radius Functions as Poisson-Delaunay Mosaics
• An intrinsic volume metric for the class of convex bodies in ${\mathbb{R}}^n$