Discrete & Computational Geometry

, Volume 13, Issue 3–4, pp 279–295

The limit shape of convex lattice polygons

  • I. Bárány


It is proved here that, asn→∞, almost all convex (1/n)ℤ2-lattice polygons lying in the square [−1, 1]2 are very close to a fixed convex set.


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  1. [A] V. I. Arnold, Statistics of integral lattice polytopes,Funksional Anal. Appl. 14 (1980), 1–3.CrossRefGoogle Scholar
  2. [BP] I. Bárány, J. Pach, On the number of convex lattice polygons,Combin. Probab. Comput. 1 (1992), 295–302.MathSciNetCrossRefMATHGoogle Scholar
  3. [BV] I. Bárány, A.M. Vershik, On the number of convex lattice polytopes,GAFA J. 2 (1992), 381–393.MathSciNetMATHGoogle Scholar
  4. [B] W. Blaschke,Vorlesungen über Differenzailgeometrie II. Affine Differenzialgeometrie, Springer-Verlag, Berlin, 1923.Google Scholar
  5. [H] G. Halász, Private communication (1993).Google Scholar
  6. [HW] Hardy, E. M. Wright,An Introduction to the Theory of Numbers, Oxford University Press, Oxford, 1938.MATHGoogle Scholar
  7. [LS] B. Logan, L. Schepp, A variational problem for random Young tableaux,Adv. in Math. 26 (1977), 206–222.CrossRefMathSciNetMATHGoogle Scholar
  8. [M] G. Meinardus, Zur additiven Zahlentheorie in mehreren Dimensionen,Math. Ann. 132 (1956), 333–346.MathSciNetCrossRefMATHGoogle Scholar
  9. [R] I. Z. Ruzsa, Private communication (1993).Google Scholar
  10. [S] Ya. G. Sinai, Probabilistic approach to analyse the statistics of convex polygonal curves,Funksional Anal. Appl. 28 (1994), 41–48 (in Russian).MathSciNetGoogle Scholar
  11. [TS] P. Turán, M. Szalay, On some problems of the statistical theory of partitions with applications to characters of the symmetric group,Acta Math. Hungar. 29 (1977), 361–379.CrossRefMATHGoogle Scholar
  12. [V] A. M. Vershik,Funksional Anal. Appl. 28 (1994), 16–25 (in Russian).MathSciNetGoogle Scholar
  13. [VK1] A. M. Vershik, S. V. Kerov, Asymptotic behavior of the Plancherel measure of the symmetric group and the limiting form of the Young tableaux,Dokl. Akad. Nauk SSSR 233 (1977), 1024–1027, (in Russian).MathSciNetMATHGoogle Scholar
  14. [VK2] A. M. Vershik, S. V. Kerov, Asymptotic theory of the characters of the symmetric group,Funksional Anal. Appl. 15 (1981), 246–255 (in Russian).MathSciNetCrossRefMATHGoogle Scholar
  15. [W] E. M. Wright, The number of partitions of a large bi-partite number,Proc. London Math. Soc. 7 (1957), 150–160.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1995

Authors and Affiliations

  • I. Bárány
    • 1
  1. 1.Mathematical Institute of the Hungarian Academy of SciencesBudapestHungary

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