Optimal Bound on the Combinatorial Complexity of Approximating Polytopes

TL;DR

This paper establishes the optimal bound on the combinatorial complexity of approximating convex bodies by polytopes in fixed dimensions, improving understanding of the minimal face count needed for accurate approximation.

Contribution

It introduces a new relationship between epsilon-width caps of a convex body and its polar, leading to an optimal polytope approximation bound in terms of combinatorial complexity.

Findings

Achieves an $O(1/ ext{epsilon}^{(d-1)/2})$ bound on the number of faces, matching the known lower bounds.

Develops a volume-sensitive bound on the number of essential caps for approximation.

Introduces a novel layered cap covering technique for efficient approximation.

Abstract

This paper considers the question of how to succinctly approximate a multidimensional convex body by a polytope. Given a convex body $K$ of unit diameter in Euclidean $d$-dimensional space (where $d$ is a constant) and an error parameter $\varepsilon > 0$, the objective is to determine a convex polytope of low combinatorial complexity whose Hausdorff distance from $K$ is at most $\varepsilon$. By combinatorial complexity we mean the total number of faces of all dimensions. Classical constructions by Dudley and Bronshteyn/Ivanov show that $O(1/\varepsilon^{(d-1)/2})$ facets or vertices are possible, respectively, but neither achieves both bounds simultaneously. In this paper, we show that it is possible to construct a polytope with $O(1/\varepsilon^{(d-1)/2})$ combinatorial complexity, which is optimal in the worst case. Our result is based on a new relationship between $\varepsilon$-width caps of a convex body and its polar body. Using this relationship, we are able to obtain a volume-sensitive bound on the number of approximating caps that are "essentially different." We achieve our main result by combining this with a variant of the witness-collector method and a novel variable-thickness layered construction of the economical cap covering.