Optimal Bound on the Combinatorial Complexity of Approximating Polytopes

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.