# Approximation of smooth convex bodies by random polytopes

## TL;DR

This paper provides an optimal upper bound for approximating smooth convex bodies in n-dimensional space by random polytopes with a fixed number of vertices, generalizing previous results and optimizing dependence on the body and density.

## Contribution

It extends prior work by establishing an optimal approximation bound for convex bodies using random polytopes with a specified vertex count, incorporating a general density function.

## Key findings

- Provides an upper bound for convex body approximation in symmetric difference metric.
- Generalizes previous results by Ludwig, Schütt, and Werner.
- Achieves optimal dependence on the number of vertices and the convex body.

## Abstract

Let $K$ be a convex body in $\mathbb{R}^n$ and $f : \partial K \rightarrow \mathbb{R}_+$ a continuous, strictly positive function with $\int\limits_{\partial K} f(x) d \mu_{\partial K}(x) = 1$. We give an upper bound for the approximation of $K$ in the symmetric difference metric by an arbitrarily positioned polytope $P_f$ in $\mathbb{R}^n$ having a fixed number of vertices. This generalizes a result by Ludwig, Sch\"utt and Werner $[36]$. The polytope $P_f$ is obtained by a random construction via a probability measure with density $f$. In our result, the dependence on the number of vertices is optimal. With the optimal density $f$, the dependence on $K$ in our result is also optimal.