# Random polytopes and the wet part for arbitrary probability distributions

## TL;DR

This paper extends classical geometric probability results to arbitrary distributions, analyzing how the convex hull's properties relate to the measure's wet part, with bounds that depend on sample size.

## Contribution

It generalizes bounds on the measure and vertices of convex hulls from uniform to arbitrary distributions, showing tightness of the bounds.

## Key findings

- Lower bounds from uniform case hold generally
- Upper bounds require a logarithmic factor adjustment
- Example demonstrates the bounds' tightness

## Abstract

We examine how the measure and the number of vertices of the convex hull of a random sample of $n$ points from an arbitrary probability measure in $\mathbf{R}^d$ relates to the wet part of that measure. This extends classical results for the uniform distribution from a convex set [B\'ar\'any and Larman 1988]. The lower bound of B\'ar\'any and Larman continues to hold in the general setting, but the upper bound must be relaxed by a factor of $\log n$. We show by an example that this is tight.