From Circles to Convex Bodies: Approximating Curved Shapes by Polytopes

TL;DR

This survey explores how smooth convex bodies can be approximated by polytopes with a limited number of faces, highlighting the universal decay rate of approximation errors across various geometric measures.

Contribution

It provides a comprehensive overview of the approximation rates of convex bodies by polytopes, including classical results, recent advances, and open problems in the field.

Findings

Approximation errors decay like N^{-2/(d-1)} for smooth convex bodies.

Random polytopes can nearly match optimal approximation quality.

The Euclidean ball serves as a benchmark for the hardest approximation case.

Abstract

Polytopes are the basic finite data structures for convex sets: they appear as feasible regions in linear optimization, as geometric summaries in algorithms, and as random objects in stochastic geometry. A natural geometric question is therefore: how well can a smooth, curved convex body be approximated by a polytope with only $N$ faces? A striking phenomenon is that in $\mathbb{R}^d$, many seemingly different approximation errors--such as volume, surface area, and others) often decay like $N^{-2/(d-1)}$ when the body has smooth, positively curved boundary. This survey article offers a guided tour of that ``universal exponent'', starting from the classical approximation of a circle by an $N$-gon and building intuition via spherical caps and curvature. We then survey a few representative theorems--including results showing that random polytopes can be almost as good as best possible ones--and explain why the Euclidean ball is a natural benchmark for the ``hardest" case. We also highlight a recently introduced projection-based distance that compares bodies through the average distance of their shadows. Finally, we list accessible open problems about sharp constants, dimension dependence, and gaps between known upper and lower bounds.