Optimal Area-Sensitive Bounds for Polytope Approximation Sunil Arya∗ Guilherme D. da Fonseca† David M. Mount‡ Department of Computer Dept. de Informática Aplicada Department of Computer Science and Engineering Universidade Federal do Science and Institute for The Hong Kong University of Estado do Rio de Janeiro Advanced Computer Studies Science and Technology (UniRio) University of Maryland Clear Water Bay, Kowloon Rio de Janeiro, Brazil College Park, Maryland 20742 Hong Kong [email protected] [email protected] [email protected] ABSTRACT Mahler volume, it is possible to achieve the desired width- Approximating convex bodies is a fundamental question in based sampling. geometry and has applications to a wide variety of optimiza- tion problems. Given a convex body K in Rd for fixed d, Categories and Subject Descriptors the objective is to minimize the number of vertices or facets F.2.2 [Analysis of Algorithms and Problem Complex- of an approximating polytope for a given Hausdorff error ity]: Nonnumerical Algorithms and Problems—Geometrical ε. The best known uniform bound, due to Dudley (1974), problems and computations shows that O((diam(K)/ε)(d−1)/2) facets suffice. While this bound is optimal in the case of a Euclidean ball, it is far General Terms from optimal for skinny convex bodies. We show that, under the assumption that the width of the Algorithms, Theory body in any direction is at least ε, it is possible to approx- imate a convex body using O( area(K)/ε(d−1)/2) facets, Keywords where area(K) is the surface area of the body. This bound Convexity, polytopes, approximation, Macbeath regions is never worse than the previousp bound and may be signifi- cantly better for skinny bodies. This bound is provably op- 1. INTRODUCTION timal in the worst case and improves upon our earlier result Approximating convex bodies by polytopes is a funda- (which appeared in SODA 2012). mental problem, which has been extensively studied in the Our improved bound arises from a novel approach to sam- literature. (See Bronstein [13] for a recent survey.) At issue pling points on the boundary of a convex body in order to is the minimum number of vertices (alternatively, the min- stab all (dual) caps of a given width. This approach in- imum number of facets) needed in an approximating poly- volves the application of an elegant concept from the theory tope for a given error ε > 0. Consider a convex body K of convex bodies, called Macbeath regions. While Macbeath in Euclidean d-dimensional space. A polytope P is said to regions are defined in terms of volume considerations, we ε-approximate K if the Hausdorff distance [13] between K show that by applying them to both the original body and and P is at most ε. Throughout, we will restrict attention its dual, and then combining this with known bounds on the to the Hausdorff metric, and we assume that the dimension d is a constant. ∗This author’s work was supported by the Research Grants Our interest is in establishing bounds on the combinato- Council, Hong Kong, China under project number 610108. rial complexity of approximating general convex bodies. Ap- †This author’s work was supported by CNPq and FAPERJ proximation bounds are of two common types. In both cases, grants. it is shown that there exists ε0 > 0 such that the bounds hold for all ε ε0. In the first type, which we call nonuni- ‡This author’s work was supported in part by the National ≤ Science Foundation under grant CCF-1117259 and the Office form bounds, the value of ε0 depends on K (for example, of Naval Research under grant N00014-08-1-1015. on K’s maximum curvature). Such bounds are often stated as holding “in the limit” as ε approaches zero, or equiva- lently as the combinatorial complexity of the approximating polytope approaches infinity. Examples include bounds by Permission to make digital or hard copies of all or part of this work for Gruber [22], Clarkson [16], and others [9,26,28,29]. personal or classroom use is granted without fee provided that copies are In the second type, which we call uniform bounds, the not made or distributed for profit or commercial advantage and that copies value of ε0 is independent of K. For example, these include bear this notice and the full citation on the first page. To copy otherwise, to the results of Dudley [19] and Bronshteyn and Ivanov [12]. republish, to post on servers or to redistribute to lists, requires prior specific These bounds hold without any smoothness assumptions. permission and/or a fee. Dudley showed that, for ε 1, any convex body K can be ε- SCG'12, June 17–20, 2012, Chapel Hill, North Carolina, USA. ≤ (d−1)/2 Copyright 2012 ACM 978-1-4503-1299-8/12/06 ...$10.00. approximated by a polytope P with O((diam(K)/ε) ) facets. Bronshteyn and Ivanov showed the same bound holds caps, we define a hyperplanar surface, which we call a base, for the number of vertices. Constants hidden in the O- whose area is less than or equal to the associate cap or dual notation depend only on d. These results have many ap- cap, respectively. Of particular interest are caps and dual plications, for example, in the construction of coresets [1]. caps whose defining width is ε. Depending on the formu- The approximation bounds of both Dudley and Bron- lation, the approximation problem reduces to computing a shteyn and Ivanov are tight up to constant factors (specif- small set of points on the boundary of K such that every ically when K is a Euclidean ball). These bounds may be cap of width ε contains one of these points or every dual significantly suboptimal if K is skinny, however. In an ear- cap of width ε contains one of these points. lier paper [3], we presented an upper bound that is based The source of slackness in the bound of [3] arises from a not on diameter, but on surface area. In particular, let sampling method that is based on a relatively heavy-handed area(K) denote the (d 1)-dimensional Hausdorff measure tool, namely ε-nets for halfspace ranges. In light of recent of ∂K. We showed that,− under the assumption that the lower bounds on the size of ε-nets for halfspace ranges [27], width of the body in any direction is at least ε, there ex- it is clear that the elimination of the log factor requires a ists an ε-approximating polytope whose number of facets is sampling process that is specially tailored to caps or dual O(t log t), where t = area(K)/ε(d−1)/2. For a given diam- caps. The principal contribution of this paper is such a eter, the surface area of a convex body is maximized for a sampling method. Euclidean ball, implyingp that area(K) = O(diam(K)d−1). Our new approach makes use of a classical structure from Thus, this bound is tight in the worst case up to the log- the theory of convexity, called Macbeath regions. Intuitively, arithmic factor. The additional log factor is disconcerting for any convex body K and a volume parameter v, there ex- since it implies that the bound is suboptimal even for the ists a collection of pairwise disjoint bodies, each of volume simple case of a Euclidean ball. In this paper we show that Ω(v), such that for every halfspace H where the cap K H the logarithmic factor can be eliminated. In particular, we has volume v, one of these bodies will be completely con-∩ prove the following result, which is worst-case optimal, up tained within this cap. (The formal statement is given in to constant factors. Section 2.4.) Macbeath regions have found numerous uses in the theory of convex sets and the geometry of numbers (see Theorem 1.1. Consider real d-space, Rd. There exists a B´ar´any [7]). To date, the application of Macbeath regions in positive ε0 and constant cd such that for any convex body the field of computational geometry has been quite limited. d K R and any ε, 0 < ε ε0, if the width of K in any For example, they have been used as a technical device in ⊂ ≤ direction is at least ε, then there exists an ε-approximating proving lower bounds for range searching (see, e.g., [4,5,10]). polytope P whose number of facets is at most Because their definition is based on volume, not width, the (d−1)/2 use of Macbeath regions in the context of uniform bounds cd area(K)/ε . for convex approximation has been limited to volume-based notions of distance, such as the Nikodym metric (which is Note that the widthp assumption seems to be a technical based on the volume of the symmetric difference) [6,8]. The necessity. For example, consider a (d 2)-dimensional unit ′ difficulty in adapting Macbeath regions to width-based sam- ball B embedded within Rd, and let B −denote its Minkowski pling is that caps of a given volume may have widely vary- sum with a d-dimensional ball of radius δ ε. By the opti- − ing widths. Our approach to dealing with this is through mality of Dudley’s bound for Euclidean balls,≪ Ω(1/ε(d 3)/2)) the application of a two-pronged strategy, which combines facets are needed to approximate B and hence to approxi- ′ ′ Macbeath-based sampling in both the original body and its mate B . But, the surface area of B can be made arbitrarily dual.