Volume approximation of convex bodies by polytopes - a constructive method

Yehoram Gordon; Mathieu Meyer; Shlomo Reisner

Studia Mathematica (1994)

  • Volume: 111, Issue: 1, page 81-95
  • ISSN: 0039-3223

Abstract

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Algorithms are given for constructing a polytope P with n vertices (facets), contained in (or containing) a given convex body K in ℝ d , so that the ratio of the volumes |K∖P|/|K| (or |P∖K|/|K|) is smaller than f ( d ) / n 2 / ( d - 1 ) .

How to cite

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Gordon, Yehoram, Meyer, Mathieu, and Reisner, Shlomo. "Volume approximation of convex bodies by polytopes - a constructive method." Studia Mathematica 111.1 (1994): 81-95. <http://eudml.org/doc/216121>.

@article{Gordon1994,
abstract = {Algorithms are given for constructing a polytope P with n vertices (facets), contained in (or containing) a given convex body K in $ℝ^d$, so that the ratio of the volumes |K∖P|/|K| (or |P∖K|/|K|) is smaller than $f(d)/n^\{2/(d-1)\}$.},
author = {Gordon, Yehoram, Meyer, Mathieu, Reisner, Shlomo},
journal = {Studia Mathematica},
keywords = {convex bodies; polytopes; approximation; polytope; symmetric difference metric; convex body},
language = {eng},
number = {1},
pages = {81-95},
title = {Volume approximation of convex bodies by polytopes - a constructive method},
url = {http://eudml.org/doc/216121},
volume = {111},
year = {1994},
}

TY - JOUR
AU - Gordon, Yehoram
AU - Meyer, Mathieu
AU - Reisner, Shlomo
TI - Volume approximation of convex bodies by polytopes - a constructive method
JO - Studia Mathematica
PY - 1994
VL - 111
IS - 1
SP - 81
EP - 95
AB - Algorithms are given for constructing a polytope P with n vertices (facets), contained in (or containing) a given convex body K in $ℝ^d$, so that the ratio of the volumes |K∖P|/|K| (or |P∖K|/|K|) is smaller than $f(d)/n^{2/(d-1)}$.
LA - eng
KW - convex bodies; polytopes; approximation; polytope; symmetric difference metric; convex body
UR - http://eudml.org/doc/216121
ER -

References

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  1. [1] U. Betke and J. M. Wills, Diophantine approximation of convex bodies, manuscript, 1979. 
  2. [2] E. M. Bronshteĭn and L. D. Ivanov, The approximation of convex sets by polyhedra, Sibirsk. Mat. Zh. 16 (1975), 1110-1112 (in Russian); English transl.: Siberian Math. J. 16 (1975), 852-853. 
  3. [3] R. Dudley, Metric entropy of some classes of sets with differentiable boundaries, J. Approx. Theory 10 (1974), 227-236. Zbl0275.41011
  4. [4] L. Fejes Tóth, Lagerungen in der Ebene, auf der Kugel und im Raum, Springer, 1953, 1972. Zbl0052.18401
  5. [5] P. M. Gruber, Approximation of convex bodies, in: Convexity and its Applications, P. M. Gruber and J. M. Wills (eds.), Birkhäuser, 1983, 131-162. 
  6. [6] P. M. Gruber, Aspects of approximation of convex bodies, in: Handbook of Convex Geometry, vol. A, P. M. Gruber and J. M. Wills (eds.), Elsevier, 1993. Zbl0791.52007
  7. [7] P. M. Gruber, Asymptotic estimates for best and stepwise approximation of convex bodies I, Forum Math. 5 (1993), 281-297. Zbl0780.52005
  8. [8] P. M. Gruber, Asymptotic estimates for best and stepwise approximation of convex bodies II, ibid., 521-538. Zbl0788.41020
  9. [9] A. M. Macbeath, An extremal property of the hypersphere, Proc. Cambridge Philos. Soc. 47 (1951), 245-247. Zbl0042.40801
  10. [10] J. S. Müller, Approximation of the ball by random polytopes, J. Approx. Theory 63 (1990), 198-209. Zbl0736.41027
  11. [11] E. Sas, Über eine Extremaleigenschaft der Ellipsen, Compositio Math. 6 (1939), 468-470. 

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