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Approximation of the sphere by polytopes having few vertices


Authors: I. Bárány and Z. Füredi
Journal: Proc. Amer. Math. Soc. 102 (1988), 651-659
MSC: Primary 52A40; Secondary 52A22
DOI: https://doi.org/10.1090/S0002-9939-1988-0928998-8
MathSciNet review: 928998
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Abstract: How well can a polytope with $ n$ vertices approximate the unit ball $ {B^d}$ of the $ d$-dimensional Euclidean space? The answer is quite well known when $ d$ is fixed and $ n$ tends to infinity. In this paper the same question is answered when $ n$ is a function of $ d$ (a polynomial in $ d$, say) and $ d$ tends to infinity. Some applications of the results are also indicated.


References [Enhancements On Off] (What's this?)

  • [1] I. Bárány and Z. Füredi, Computing the volume is difficult, Discrete and Computational Geometry 2 (1987), 319-326 and Proc. 18th ACM STOC, 1986, pp. 442-447. MR 911186 (89a:68203)
  • [2] -, On the convex hull of random points, Probability Theory and Related Fields (to appear).
  • [3] C. Buchta, J. Müller and R. F. Tichy, Stochastical approximation of convex bodies, Math. Ann. 271 (1985), 225-235. MR 783553 (86g:52009)
  • [4] L. Danzer, B. Grünbaum and V. Klee, Helly's theorem and its relatives, Proc. Sympos. Pure Math., vol. 7, Amer. Math. Soc., Providence, R. I., 1963, pp. 101-180.
  • [5] R. Dudley, Metric entropy of some classes of sets with differentiable boundaries, J. Approx. Theory 10 (1974), 227-236; Correction, ibid. 26 (1979), 192-193. MR 0358168 (50:10633)
  • [6] G. Elekes, A geometric inequality and the complexity of computing the volume, Discrete and Computational Geometry 1 (1986), 289-292. MR 866364 (87k:68138)
  • [7] L. Fejes-Tóth, Regular figures, Pergamon Press, 1964. MR 0165423 (29:2705)
  • [8] P. M. Gruber, Approximation of convex bodies, Convexity and its Applications (P. M. Gruber and J. M. Wills, eds.), Birkhäuser, 1983, pp. 131-162. MR 731110 (85d:52001)
  • [9] P. M. Gruber and P. Kendarov, Approximation of convex bodies by polytopes, Rend. Circ. Mat. Palermo 31 (1982), 195-225. MR 670396 (84d:52004)
  • [10] L. Lovász, An algorithmic theory of numbers, graphs and convexity, Report No. 85368-OR, Univ. Bonn, 1985.
  • [11] A. M. Macbeath, An extremal property of the hypersphere, Proc. Cambridge Philos. Soc. 47 (1951), 245-247. MR 0039292 (12:526e)
  • [12] R. Schneider, Zur optimalen Approximation konvexer Hyperflächen durch Polyeder, Math. Ann. 256 (1981), 289-301. MR 626950 (82m:52003)
  • [13] R. Schneider and J. A. Wieacker, Approximation of convex bodies by polytopes, Bull. London Math. Soc. 13 (1981), 149-156. MR 608101 (82g:52008)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1988-0928998-8
Article copyright: © Copyright 1988 American Mathematical Society

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