# On the geometry of random polytopes

## TL;DR

This paper provides a simple proof of a recent result showing that the convex hull of certain random matrix rows approximates a specific geometric body under minimal assumptions.

## Contribution

It offers a straightforward proof of a geometric approximation result for random polytopes generated by symmetric random matrices.

## Key findings

- Convex hull of random matrix rows approximates a specific geometric body.
- High probability bounds for the approximation.
- Minimal assumptions on the distribution of matrix entries.

## Abstract

We present a simple proof to a fact recently established in [5]: let $\xi$ be a symmetric random variable that has variance $1$, let $\Gamma=(\xi_{ij})$ be an $N \times n$ random matrix whose entries are independent copies of $\xi$, and set $X_1,...,X_N$ to be the rows of $\Gamma$. Then under minimal assumptions on $\xi$ and as long as $N \geq c_1n$, $$ c_2 \bigl(B_\infty^n \cap \sqrt{\log(eN/n)} B_2^n \bigr) \subset {\rm absconv}(X_1,...,X_N) $$ with high probability.