Optimal Bound on the Combinatorial Complexity of Approximating Polytopes

Abstract

Convex bodies play a fundamental role in geometric computation, and approximating such bodies is often a key ingredient in the design of efficient algorithms. We consider the question of how to succinctly approximate a multidimensional convex body by a polytope. We are given a convex body K of unit diameter in Euclidean d-dimensional space (where d is a constant) along with an error parameter ε > 0. The objective is to determine a polytope of low combinatorial complexity whose Hausdorff distance from K is at most ε. By combinatorial complexity we mean the total number of faces of all dimensions of the polytope. In the mid-1970's, a result by Dudley showed that O(1/ε (d−1)/2 ) facets suffice, and Bronshteyn and Ivanov presented a similar bound on the number of vertices. While both results match known worst-case lower bounds, obtaining a similar upper bound on the total combinatorial complexity has been open for over 40 years. Recently, we made a first step forward towards this objective, obtaining a suboptimal bound. In this paper, we settle this problem with an asymptotically optimal bound of O(1/ε (d−1)/2 ).Our result is based on a new relationship between ε-width caps of a convex body and its polar. 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 result by combining this with a variant of the witness-collector method and a novel variable-width layered construction.

Voronoi Diagrams in the Hilbert Metric

“…We came to consider the Hilbert geometry because of its relevance to the topic of convex approximation. Efficient approximations of convex bodies have been applied to a wide range of applications, including approximate nearest neighbor searching both in Euclidean space [6] and more general metrics [1], optimal construction of ε-kernels [4], solving the closest vector problem approximately [9,10,14,19], computing approximating polytopes with low combinatorial complexity [3,5]. These works all share one thing in common-they approximate a convex body by covering it with elements that behave much like metric balls.…”

Uniform Brackets, Containers, and Combinatorial Macbeath Regions

“…Since its introduction Macbeath regions have been an important object of study in convex geometry [B 00, B 08]. More recently, Macbeath regions were used for proving data structure lower bounds [BCP93,AMX12], and convex body approximation problems in computational geometry [AdFM17a,AdFM17b,AAdFM20]. Mnets were proposed as combinatorial analogues of Macbeath's theorem by Mustafa and Ray [MR17], who showed their existence for many geometrically defined classes of set systems.…”