Computing the volume is difficult

@article{Brny1986ComputingTV,
  title={Computing the volume is difficult},
  author={Imre B{\'a}r{\'a}ny and Zolt{\'a}n F{\"u}redi},
  journal={Discrete \& Computational Geometry},
  year={1986},
  volume={2},
  pages={319-326}
}
AbstractFor every polynomial time algorithm which gives an upper bound % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVC0xe9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Gqpi0x% c9q8qqaqFj0df9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaie% aacaWF2bGaa83Baiaa-Xgaaaaaaa!3A2B! $$\overline {vol}$$ (K) and a lower boundvol(K) for the volume of a convex setK⊂Rd, the… 

Approximating the volume of convex bodies

AbstractIt is a well-known fact that for every polynomial-time algorithm which gives an upper bound % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn%

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References

SHOWING 1-10 OF 34 REFERENCES

A geometric inequality and the complexity of computing volume

  • G. Elekes
  • Mathematics
    Discret. Comput. Geom.
  • 1986
The volume of the convex hull of anym points of ann-dimensional ball with volumeV is at mostV·m/2n. This implies that no polynomial time algorithm can compute the volume of a convex set (given by an

Algorithmic theory of numbers, graphs and convexity

  • L. Lovász
  • Mathematics
    CBMS-NSF regional conference series in applied mathematics
  • 1986
TLDR
How to Round Numbers Preliminaries and some Applications in Combinatorics Cuts and Joins Chromatic Number, Cliques and Perfect Graphs Minimizing a Submodular Function.

An extremal property of the hypersphere

  • A. Macbeath
  • Mathematics
    Mathematical Proceedings of the Cambridge Philosophical Society
  • 1951
It was shown by Sas (1) that, if K is a plane convex body, then it is possible to inscribe in K a convex n-gon occupying no less a fraction of its area than the regular n-gon occupies in its

Stochastical approximation of convex bodies

Enveloppe convexe de points aleatoires sur la sphere. Enveloppe convexe de points aleatoires sur la frontiere d'un corps convexe lisse

Problem

Philosophical and ideological approach to the study of childhood as a social phenomenon is substantiated in the article. The meaning of human existence and his relationship with the surrounding

An algorithmic theory of numbers, graphs and convexity, Preprint

  • Report No. 85368-OR,
  • 1985

Sections euclidiennes et volume des corps symetriques convexes darts R

  • C. R. Acad. Sci. Paris Sdr. I
  • 1985

Approximation ofthe ball by polytopes having few vertices

    Sections euclidiennes et volume des corps symetriques convexes dans

    • C. R. Acad. Sci. Paris Sár
    • 1985