# Fine approximation of convex bodies by polytopes

## TL;DR

This paper proves that any convex body can be closely approximated by a polytope with a controlled number of vertices, providing bounds that depend exponentially on dimension and polynomially on the approximation parameter.

## Contribution

It establishes a new bound on the number of vertices needed for polytope approximation of convex bodies in high dimensions.

## Key findings

- Existence of polytopes with controlled vertices approximating convex bodies
- Bounds depend exponentially on dimension and polynomially on approximation accuracy
- Approximation preserves the convex body's shape within a factor of (1 - epsilon)

## Abstract

We prove that for every convex body $K$ with the center of mass at the origin and every $\varepsilon\in \left(0,\frac{1}{2}\right)$, there exists a convex polytope $P$ with at most $e^{O(d)}\varepsilon^{-\frac{d-1}{2}}$ vertices such that $(1-\varepsilon)K\subset P\subset K$.