Discrete & Computational Geometry

, Volume 2, Issue 4, pp 319–326 | Cite as

Computing the volume is difficult

  • Imre Bárány
  • Zoltán Füredi


For every polynomial time algorithm which gives an upper bound\(\overline {vol}\)(K) and a lower boundvol(K) for the volume of a convex setKR d , the ratio\(\overline {vol}\)(K)/vol(K) is at least (cd/logd) d for some convex setKR d .


Convex Hull Convex Body Polynomial Time Algorithm Discrete Comput Geom Symmetric Convex Body 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    I. Bárány and Z. Füredi, Approximation of the ball by polytopes having few vertices, to appear.Google Scholar
  2. 2.
    J. Bourgain and V. D. Milman, Sections euclidiennes et volume des corps symetriques convexes dansR n,C. R. Acad. Sci. Paris Sér. I 300 (1985), 435–437.MathSciNetzbMATHGoogle Scholar
  3. 3.
    C. Buchta, J. Müller, and R. F. Tichy, Stochastical approximation of convex bodies,Math. Ann. 271 (1985), 225–235.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    L. Danzer, B. Grünbaum, and V. Klee, Helly's theorem and its relatives, inProceedings of Symposia in Pure Math, Vol. VII (V. Klee, ed.), Providence, RI, 1983.Google Scholar
  5. 5.
    G. Elekes, A geometric inequality and the complexity of measuring the volume,Discrete Comput. Geom., to appear.Google Scholar
  6. 6.
    L. Fejes Tóth,Regular Figures, Pergamon Press, Oxford, 1964.zbMATHGoogle Scholar
  7. 7.
    M. Grötschel, L. Lovász, and A. Schrijver,Combinatorial Optimization and the Ellipsoid Method, Springer-Verlag, New York, 1987.Google Scholar
  8. 8.
    L. Lovász, private communication, 1983.Google Scholar
  9. 9.
    L. Lovász, An algorithmic theory of numbers, graphs and convexity, Preprint, Report No. 85368-OR, University of Bonn, 1985.Google Scholar
  10. 10.
    A. M. Macbeath, An extremal property of the hypersphere,Math. Proc. Cambridge Phil. Soc. 47 (1951), 245–247.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    W. O. J. Moser and J. Pach,Research Problems in Discrete Geometry, Problem 76, Montreal, 1985.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1987

Authors and Affiliations

  • Imre Bárány
    • 1
  • Zoltán Füredi
    • 2
  1. 1.Department of Operations Research and Industrial EngineeringCornell UniversityIthacaUSA
  2. 2.RUTCORRutgers UniversityNew BrunswickUSA

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