# Asymptotic normality for random simplices and convex bodies in high dimensions

## TL;DR

This paper proves central limit theorems for the log-volume of high-dimensional random convex bodies, including simplices and bodies generated from radially symmetric measures, showing asymptotic normality as dimension grows.

## Contribution

It establishes the asymptotic normality of log-volumes for a broad class of high-dimensional random convex bodies, extending previous results to new distributions and settings.

## Key findings

- Asymptotic normality for log-volumes of random simplices with i.i.d. vertices.
- Normal approximation for convex bodies with vertices from radially symmetric measures.
- Results hold as dimension tends to infinity, covering various distributions.

## Abstract

Central limit theorems for the log-volume of a class of random convex bodies in $\mathbb{R}^n$ are obtained in the high-dimensional regime, that is, as $n\to\infty$. In particular, the case of random simplices pinned at the origin and simplices where all vertices are generated at random is investigated. The coordinates of the generating vectors are assumed to be independent and identically distributed with subexponential tails. In addition, asymptotic normality is established also for random convex bodies (including random simplices pinned at the origin) when the spanning vectors are distributed according to a radially symmetric probability measure on the $n$-dimensional $\ell_p$-ball. In particular, this includes the cone and the uniform probability measure.