# Limit theorems for random polytopes with vertices on convex surfaces

## TL;DR

This paper establishes limit theorems, variance bounds, and normal approximation results for the intrinsic volumes of random polytopes formed by points on convex surfaces, advancing understanding of their probabilistic geometric properties.

## Contribution

It provides new variance bounds, strong laws, and central limit theorems for intrinsic volumes of convex hulls of boundary points, using Stein's method and surface body estimates.

## Key findings

- Variance bounds for intrinsic volumes
- Strong laws of large numbers
- Central limit theorems with normal approximation

## Abstract

The random polytope $K_n$, defined as the convex hull of $n$ points chosen uniformly at random on the boundary of a smooth convex body, is considered. Proofs for lower and upper variance bounds, strong laws of large numbers and central limit theorems for the intrinsic volumes of $K_n$ are presented. A normal approximation bound from Stein's method and estimates for surface bodies are among the involved tools.