Volume approximation of convex bodies by polytopes - a constructive method

• Autorzy: Yehoram Gordon , Mathieu Meyer , Shlomo Reisner
• Języki publikacji: EN

Abstrakty

EN

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 $f(d)/n^{2/(d-1)}$.

Słowa kluczowe

EN

convex bodies; polytopes; approximation

Kategorie tematyczne

• 52A20: Convex sets in n dimensions (including convex hypersurfaces)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1994
• Tom: 111
• Numer: 1
• Strony: 81-95

Daty

wydano

1994

otrzymano

1993-11-12

poprawiono

1994-01-26

Twórcy

autor

Yehoram Gordon

• Department of Mathematics, Technion I.I.T., 32000 Haifa, Israel

autor

Mathieu Meyer

• Equipe d'Analyse, Université Paris 6, 4, Place Jussieu, F-75252 Paris Cedex 5, France

autor

Shlomo Reisner

• Department of Mathematics and School of Education - Oranim, University of Haifa 31905 Haifa, Israel

Bibliografia

• [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.
• [4] L. Fejes Tóth, Lagerungen in der Ebene, auf der Kugel und im Raum, Springer, 1953, 1972.
• [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.
• [7] P. M. Gruber, Asymptotic estimates for best and stepwise approximation of convex bodies I, Forum Math. 5 (1993), 281-297.
• [8] P. M. Gruber, Asymptotic estimates for best and stepwise approximation of convex bodies II, ibid., 521-538.
• [9] A. M. Macbeath, An extremal property of the hypersphere, Proc. Cambridge Philos. Soc. 47 (1951), 245-247.
• [10] J. S. Müller, Approximation of the ball by random polytopes, J. Approx. Theory 63 (1990), 198-209.
• [11] E. Sas, Über eine Extremaleigenschaft der Ellipsen, Compositio Math. 6 (1939), 468-470.