Intrinsic volumes of inscribed random polytopes in smooth convex bodies

TL;DR

This paper investigates the variability and convergence of intrinsic volumes of random polytopes inscribed in smooth convex bodies, providing bounds and laws of large numbers using geometric and probabilistic tools.

Contribution

It establishes matching bounds for the variances and proves strong laws of large numbers for intrinsic volumes of inscribed random polytopes in smooth convex bodies.

Findings

Derived bounds for variances of intrinsic volumes.

Proved strong laws of large numbers for intrinsic volumes.

Used geometric covering and probabilistic inequalities as key tools.

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 B\'ar\'any and Larman, and the Efron-Stein jackknife inequality.