A limit theorem for Hausdorff approximation by random inscribed polytopes

TL;DR

This paper establishes a limit theorem showing that the Hausdorff approximation error, when approximating a smooth convex body with random inscribed polytopes, converges to a Gumbel distribution under optimal sampling.

Contribution

It provides a new probabilistic limit theorem for the Hausdorff approximation error using random inscribed polytopes and connects it to covering properties and extreme value theory.

Findings

Rescaled approximation error converges to Gumbel distribution.

Optimal boundary sampling density is crucial for the limit theorem.

The proof links geometric covering properties to probabilistic limit laws.

Abstract

Approximate a smooth convex body $K$ with nonvanishing curvature by the convex hull of $n$ independent random points sampled from its boundary $\partial K$. In case the points are distributed according to the optimal density, we prove that the rescaled approximation error in Hausdorff distance tends to a Gumbel distributed random variable. The proof is based on an asymptotic relation to covering properties of random geodesic balls on $\partial K$ and on a limit theorem due to Janson.