Imre Bárány

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Imre Bárány

Imre Bárány (born December 7, 1947 in Mátyásföld , Budapest ) is a Hungarian mathematician who deals with combinatorics and discrete geometry.

Bárány is a mathematician at the Alfred Renyi Institute of the Hungarian Academy of Sciences . He is also a professor at University College London.

In 1978 he gave a new, short proof of Martin Kneser's conjecture about the chromatic number of Kneser graphs. In 1980 he gave a new proof of Borsuk and Ulam's theorem . In 1981, together with SB Shlosman and A. Szucs, he proved a topological generalization of a theorem by Helge Tverberg (see Topological Combinatorics ).

Together with Zoltán Füredi he gave an algorithm for the cryptographic protocol Mental Poker and proved that the computation of the volume of a convex set in d-dimensional space defined by a membership oracle for points is a generally difficult (non-polynomial-temporal) problem .

In 2000 he solved James Joseph Sylvester's problem about the probability that randomly distributed points are in a convex position. Sylvester originally asked in 1864 about the probability that four points chosen at random in the plane form a non-convex quadrilateral. The generalization asks about the probability p (K, n) that n randomly chosen points of a convex polytope K in d dimensions are in a convex position, i.e. none of the n randomly chosen points lies in the convex hull of the other. Barany dealt with several cases of the generalized problem.

With Vershik and Pach he solved a problem by Vladimir Arnold about the number of convex polytopes made from grid points. With Van Vu he proved a central limit theorem for random polytopes.

In 1989, together with László Lovász and Füredi , he proved an asymptotic estimate for the number of planes that divide a set S of n points in three-dimensional Euclidean space in a general position into two halves (the planes each passing through three points of S). With Füredi and J. Pach he proved the six-circle conjecture of László Fejes Tóth . It says that with a circle packing in the plane in which each circle has six neighboring circles, either the hexagonal circle packing with circles of the same radius is present or circles with arbitrarily small radius occur.

In 1985 he received the Mathematics Prize (now Paul Erdős Prize ) of the Hungarian Academy of Sciences and in 2010 he became its corresponding member. In 2002 he was invited speaker at the International Congress of Mathematicians in Beijing ( Random Points, convex bodies, and lattices ). He is a fellow of the American Mathematical Society . In 2016 he was awarded the Széchenyi Prize .

Web links

Individual evidence

  1. ↑ A short proof of Knesers conjecture , J. Comb. Theory, A, Volume 25, 1978, pp. 325–326, also presented in Aigner, Ziegler: The Book of Proofs , Springer. The first proof came from László Lovász in 1978.
  2. ↑ Borsuk's theorem through complementary pivoting , Math. Programming, Volume 18, 1984, pp. 84-88. Shown in Jiri Matousek: Using the Borsuk-Ulam Theorem , Springer 2003
  3. ↑ Barany, SB Shlosman, A. Szucs: On a topological generalization of a theorem of Tverberg , J. London Math. Soc. (2), Vol. 23, 1981, pp. 158-164
  4. ↑ Barany, Füredi: Mental poker with three or more players , Information and Control, Volume 59, 1983, pp. 84-93
  5. ↑ Bárány, Füredi: Computing the volume is difficult , Discrete and Computational Geometry, Volume 2, 1987, pp. 319–326
  6. ↑ Sylvesters question: the probability that n points are in convex position , Annals of probability, Volume 27, 2000, pp. 2020-2034
  7. ↑ reentrant quadrilateral , that is, the fourth point lies within the triangle formed by the three other points.
  8. ↑ Sylvester's problem asks for the complement of this probability, that is, for the case that the points are not in a convex position
  9. ↑ see the review article by Barany: Random points and lattice points in convex bodies , Bulletin AMS, Volume 45, 2008, p. 339
  10. ↑ Barany, Pach: On the number of convex lattice polytopes, Comb. Prob. Comp., Vol. 1, 1991, p. 295, Barany, Vershik: On the number of convex lattice polytopes, Geometry and Functional Analysis, Vol. 12, 1992, p. 381
  11. ↑ Barany, Vu Central limit theorems for Gaussian polytopes , Annals of Probability, Volume 35, 2007, pp. 1593–1621 ( Memento of the original from December 20, 2016 in the Internet Archive ) Info: The archive link has been inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice.  @1@ 2Template: Webachiv / IABot / projecteuclid.org
  12. ↑ Barany, Füredi, Lovasz: On the number of halving planes , Combinatorica, Volume 10, 1990, pp. 175-185
  13. ↑ Bárány, Füredi, Pach: Discrete convex functions and proof of the six circle conjecture of L. Fejes Toth , Canadian J. Mathematics, Volume 36, 1983, pp. 569-576
personal data
SURNAME Bárány, Imre
BRIEF DESCRIPTION Hungarian mathematician
DATE OF BIRTH December 7, 1947
PLACE OF BIRTH Mátyásföld , Budapest