# 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

## Access Full Article

top## Abstract

top## How to cite

topGordon, 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

top- [1] U. Betke and J. M. Wills, Diophantine approximation of convex bodies, manuscript, 1979.
- [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] R. Dudley, Metric entropy of some classes of sets with differentiable boundaries, J. Approx. Theory 10 (1974), 227-236. Zbl0275.41011
- [4] L. Fejes Tóth, Lagerungen in der Ebene, auf der Kugel und im Raum, Springer, 1953, 1972. Zbl0052.18401
- [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] 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] P. M. Gruber, Asymptotic estimates for best and stepwise approximation of convex bodies I, Forum Math. 5 (1993), 281-297. Zbl0780.52005
- [8] P. M. Gruber, Asymptotic estimates for best and stepwise approximation of convex bodies II, ibid., 521-538. Zbl0788.41020
- [9] A. M. Macbeath, An extremal property of the hypersphere, Proc. Cambridge Philos. Soc. 47 (1951), 245-247. Zbl0042.40801
- [10] J. S. Müller, Approximation of the ball by random polytopes, J. Approx. Theory 63 (1990), 198-209. Zbl0736.41027
- [11] E. Sas, Über eine Extremaleigenschaft der Ellipsen, Compositio Math. 6 (1939), 468-470.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.