Compact convex sets of the plane and probability theory

Marckert, Jean-François; Renault, David

ESAIM: Probability and Statistics, Tome 18 (2014), pp. 854-880

The Gauss-Minkowski correspondence in ℝ² states the existence of a homeomorphism between the probability measures μ on [0,2π] such that ${\int }_0^{2\pi }{\mathrm{e}}^{ix}\mathrm{d}\mu \left(x\right)=0$ ∫ 0 2 π e ix d μ ( x ) = 0 and the compact convex sets (CCS) of the plane with perimeter 1. In this article, we bring out explicit formulas relating the border of a CCS to its probability measure. As a consequence, we show that some natural operations on CCS - for example, the Minkowski sum - have natural translations in terms of probability measure operations, and reciprocally, the convolution of measures translates into a new notion of convolution of CCS. Additionally, we give a proof that a polygonal curve associated with a sample of n random variables (satisfying ${\int }_0^{2\pi }{\mathrm{e}}^{ix}\mathrm{d}\mu \left(x\right)=0$ ∫ 0 2 π e ix d μ ( x ) = 0 ) converges to a CCS associated with μ at speed √n, a result much similar to the convergence of the empirical process in statistics. Finally, we employ this correspondence to present models of smooth random CCS and simulations.

Classification : 52A10, 60B05, 60D05, 60F17, 60G99
Keywords: random convex sets, symmetrisation, weak convergence, Minkowski sum

[1] I. Bárány, Sylvester's question: The probability that n points are in convex position. Ann. Probab. 27 (1999) 2020-2034. | Zbl | MR

[2] I. Bárány, Random polytopes, convex bodies and approximation, in Stochastic Geometry, Vol. 1892 of Lect. Notes Math. Springer Berlin/Heidelberg (2007) 77-118. | Zbl | MR

[3] I. Bárány and A.M. Vershik, On the number of convex lattice polytopes. Geom. Func. Anal. 2 (1992) 381-393. | Zbl | MR

[4] P. Billingsley, Convergence of probability measures. Wiley Series in Probability and Statistics: Probability and Statistics, 2nd edition. A Wiley-Interscience Publication. John Wiley & Sons Inc., New York (1999). | Zbl | MR

[5] O. Bodini, Ph. Duchon, A. Jacquot and L. Mutafchiev, Asymptotic analysis and random sampling of digitally convex polyominoes. In Proc. of the 17th IAPR international conference on Discrete Geometry for Computer Imagery, DGCI'13. Springer-Verlag, Berlin, Heidelberg (2013) 95-106. | MR

[6] L.V. Bogachev and S.M. Zarbaliev, Universality of the limit shape of convex lattice polygonal lines. Ann. Probab. 39 (1992) 2271-2317. | Zbl | MR

[7] C. Buchta, On the boundray structure of the convex hull of random points. Adv. Geom. (2012). Available at: http://www.uni-salzburg.at/pls/portal/docs/1/1739190.PDF. | Zbl | MR

[8] H. Busemann, Convex Surfaces. Interscience. New York (1958). | Zbl | MR

[9] P. Calka, Precise formulae for the distributions of the principal geometric characteristics of the typical cells of a two-dimensional poisson-voronoi tessellation and a poisson line process. Adv. Appl. Probab. 35 (2003) 551-562. Available at http://www.univ-rouen.fr/LMRS/Persopage/Calka/publications.html. | Zbl | MR

[10] R.M. Dudley, Real Analysis and Probability. Cambridge Studies in Advanced Mathematics. Cambridge University Press (2002). | Zbl | MR

[11] W. Feller, An introduction to probability theory and its applications. Vol. II. 2nd edition. John Wiley & Sons Inc., New York (1971). | Zbl | MR

[12] M.A. Hurwitz, Sur le problème des isopérimètres. C. R. Acad. Sci. Paris 132 (1901) 401-403. | JFM

[13] M.A. Hurwitz, Sur quelques applications géométriques des séries de Fourier. Annales Scientifiques de l'École Normale supérieure, 19 (1902) 357-408. Available at http://archive.numdam.org/article/ASENS˙1902˙3˙19˙˙357˙0.pdf. | JFM | Numdam

[14] B. Klartag, On John-type ellipsoids, in Geometric aspects of functional analysis, vol. 1850 of Lect. Notes Math. Springer, Berlin (2004) 149-158. | Zbl | MR

[15] D.E. Knuth, Axioms and hulls. Vol. 606 of Lect. Notes Comput. Sci. Springer-Verlag, Berlin (1992). Available at: http://www-cs-faculty.stanford.edu/˜uno/aah.html. | Zbl | MR

[16] P. Lévy, L'addition des variables aléatoires définies sur un circonférence. Bull. Soc. Math. France 67 (1939) 1-41. Available at http://archive.numdam.org/article/BSMF˙1939˙˙67˙˙1˙0.pdf. | Zbl | JFM | Numdam

[17] J.-F. Marckert, Probability that n random points in a disk are in convex position. Available at http://arxiv.org/abs/1402.3512 (2014).

[18] M. Moszyńska, Selected Topics in Convex Geometry. Birkhäuser (2006). | Zbl

[19] V.V. Petrov, Sums of independent random variables. Translated from the Russian by A.A. Brown. Band 82, Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag, New York (1975). | Zbl | MR

[20] A.V. Pogorelov, Extrinsic geometry of convex surfaces. American Mathematical Society, Providence, R.I. (1973). Translated from the Russian by Israel Program for Scientific Translations, in vol. 35 Translations of Mathematical Monographs. | Zbl | MR

[21] G. Pólya, Isoperimetric Inequalities in Mathematical Physics. Ann. Math. Stud. Kraus (1965). | Zbl

[22] W. Rudin, Real and Complex Analysis, 3rd edn. McGraw-Hill International Editions (1987). | Zbl | MR

[23] R. Schneider, Convex Bodies: The Brunn−Minkowski Theory. Cambridge University Press (1993). | Zbl | MR

[24] Ya.G. Sinai, Probabilistic approach to the analysis of statistics for convex polygonal lines. Functional Anal. Appl. 28 (1994) 1. | Zbl | MR

[25] J.J. Sylvester, On a special class of questions on the theory of probabilities. Birmingham British Assoc. Rept. (1865) 8-9.

[26] G. Szegö, Orthogonal polynomials. Colloquium Publications, 4th edition. American Mathematical Society (1939). | JFM

[27] P. Valtr, Probability that n random points are in convex position. Discr. Comput. Geom. 13 (1995) 637-643. | Zbl | MR

[28] P. Valtr, The probability that n random points in a triangle are in convex position. Combinatorica 16 (1996) 567-573. | Zbl | MR

[29] A. Vershik and O. Zeitouni, large deviations in the geometry of convex lattice polygons. Israel J. Math. 109 (1999) 13-27. | Zbl | MR

[30] R.J.G. Wilms, Fractional parts of random variables. Limit theorems and infinite divisibility, Dissertation. Technische Universiteit Eindhoven, Eindhoven (1994). | Zbl | MR