# Approximating a convex body by a polytope using the epsilon-net theorem

## TL;DR

This paper extends previous results by showing that a polytope approximating a convex body can be constructed from a specific number of random points using the epsilon-net theorem, with high probability.

## Contribution

It provides a new probabilistic bound for approximating convex bodies with polytopes, generalizing prior work and simplifying the proof with combinatorial tools.

## Key findings

- Number of points needed depends on dimension and approximation parameter
- High probability of successful approximation with the specified number of points
- Simplified proof using epsilon-net theorem

## Abstract

Giving a joint generalization of a result of Brazitikos, Chasapis and Hioni and results of Giannopoulos and Milman, we prove that roughly $\left\lceil \frac{d}{(1-\vartheta)^d}\ln\frac{1}{(1-\vartheta)^d} \right\rceil$ points chosen uniformly and independently from a centered convex body $K$ in ${\mathbb R}^d$ yield a polytope $P$ for which $\vartheta K\subseteq P\subseteq K$ holds with large probability. The proof is simple, and relies on a combinatorial tool, the $\varepsilon$-net theorem.