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On the Combinatorial Complexity of Approximating Polytopes

Authors Sunil Arya, Guilherme D. da Fonseca, David M. Mount

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Sunil Arya
Guilherme D. da Fonseca
David M. Mount

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Sunil Arya, Guilherme D. da Fonseca, and David M. Mount. On the Combinatorial Complexity of Approximating Polytopes. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 11:1-11:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)


Approximating convex bodies succinctly by convex polytopes is a fundamental problem in discrete geometry. A convex body K of diameter $diam(K)$ is given in Euclidean d-dimensional space, where $d$ is a constant. Given an error parameter eps > 0, the objective is to determine a polytope of minimum combinatorial complexity whose Hausdorff distance from K is at most eps diam(K). By combinatorial complexity we mean the total number of faces of all dimensions of the polytope. A well-known result by Dudley implies that O(1/eps^{(d-1)/2}) facets suffice, and a dual result by Bronshteyn and Ivanov similarly bounds the number of vertices, but neither result bounds the total combinatorial complexity. We show that there exists an approximating polytope whose total combinatorial complexity is O-tilde(1/eps^{(d-1)/2}), where O-tilde conceals a polylogarithmic factor in 1/eps. This is an improvement upon the best known bound, which is roughly O(1/eps^{d-2}). Our result is based on a novel combination of both new and old ideas. First, we employ Macbeath regions, a classical structure from the theory of convexity. The construction of our approximating polytope employs a new stratified placement of these regions. Second, in order to analyze the combinatorial complexity of the approximating polytope, we present a tight analysis of a width-based variant of Barany and Larman's economical cap covering, which may be of independent interest. Finally, we use a deterministic variation of the witness-collector technique (developed recently by Devillers et al.) in the context of our stratified construction.
  • Polytope approximation
  • Convex polytopes
  • Macbeath regions


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  1. P. K. Agarwal, S. Har-Peled, and K. R. Varadarajan. Approximating extent measures of points. J. Assoc. Comput. Mach., 51:606-635, 2004. Google Scholar
  2. G. E. Andrews. A lower bound for the volumes of strictly convex bodies with many boundary points. Trans. Amer. Math. Soc., 106:270-279, 1963. Google Scholar
  3. S. Arya, G. D. da Fonseca, and D. M. Mount. Optimal area-sensitive bounds for polytope approximation. In Proc. 28th Annu. Sympos. Comput. Geom., pages 363-372, 2012. Google Scholar
  4. S. Arya, T. Malamatos, and D. M. Mount. The effect of corners on the complexity of approximate range searching. Discrete Comput. Geom., 41:398-443, 2009. Google Scholar
  5. S. Arya, D. M. Mount, and J. Xia. Tight lower bounds for halfspace range searching. Discrete Comput. Geom., 47:711-730, 2012. URL:
  6. I. Bárány. Intrinsic volumes and f-vectors of random polytopes. Math. Ann., 285:671-699, 1989. Google Scholar
  7. I. Bárány. The technique of M-regions and cap-coverings: A survey. Rend. Circ. Mat. Palermo, 65:21-38, 2000. Google Scholar
  8. I. Bárány. Extremal problems for convex lattice polytopes: A survey. Contemp. Math., 453:87-103, 2008. Google Scholar
  9. I. Bárány and D. G. Larman. Convex bodies, economic cap coverings, random polytopes. Mathematika, 35:274-291, 1988. Google Scholar
  10. K. Böröczky, Jr. Approximation of general smooth convex bodies. Adv. Math., 153:325-341, 2000. Google Scholar
  11. H. Brönnimann, B. Chazelle, and J. Pach. How hard is halfspace range searching. Discrete Comput. Geom., 10:143-155, 1993. Google Scholar
  12. E. M. Bronshteyn and L. D. Ivanov. The approximation of convex sets by polyhedra. Siberian Math. J., 16:852-853, 1976. Google Scholar
  13. E. M. Bronstein. Approximation of convex sets by polytopes. J. Math. Sci., 153(6):727-762, 2008. Google Scholar
  14. K. L. Clarkson. Building triangulations using ε-nets. In Proc. 38th Annu. ACM Sympos. Theory Comput., pages 326-335, 2006. Google Scholar
  15. O. Devillers, M. Glisse, and X. Goaoc. Complexity analysis of random geometric structures made simpler. In Proc. 29th Annu. Sympos. Comput. Geom., pages 167-176, 2013. Google Scholar
  16. R. M. Dudley. Metric entropy of some classes of sets with differentiable boundaries. J. Approx. Theory, 10(3):227-236, 1974. Google Scholar
  17. G. Ewald, D. G. Larman, and C. A. Rogers. The directions of the line segments and of the r-dimensional balls on the boundary of a convex body in Euclidean space. Mathematika, 17:1-20, 1970. Google Scholar
  18. P. M. Gruber. Asymptotic estimates for best and stepwise approximation of convex bodies I. Forum Math., 5:521-537, 1993. Google Scholar
  19. S. Har-Peled. Geometric approximation algorithms. Number 173 in Mathematical surveys and monographs. American Mathematical Society, 2011. Google Scholar
  20. F. John. Extremum problems with inequalities as subsidiary conditions. In Studies and Essays Presented to R. Courant on his 60th Birthday, pages 187-204. Interscience Publishers, Inc., New York, 1948. Google Scholar
  21. A. M. Macbeath. A theorem on non-homogeneous lattices. Ann. of Math., 56:269-293, 1952. Google Scholar
  22. P. McMullen. The maximum numbers of faces of a convex polytope. Mathematika, 17:179-184, 1970. Google Scholar
  23. R. Schneider. Polyhedral approximation of smooth convex bodies. J. Math. Anal. Appl., 128:470-474, 1987. Google Scholar
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