The Gauss-Minkowski correspondence in ℝ2 states the existence of a homeomorphism between the probability measures μ on [0,2π] such that ∫ 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 ∫ 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.
Mots-clés : random convex sets, symmetrisation, weak convergence, Minkowski sum
@article{PS_2014__18__854_0, author = {Marckert, Jean-Fran\c{c}ois and Renault, David}, title = {Compact convex sets of the plane and probability theory}, journal = {ESAIM: Probability and Statistics}, pages = {854--880}, publisher = {EDP-Sciences}, volume = {18}, year = {2014}, doi = {10.1051/ps/2014008}, language = {en}, url = {http://www.numdam.org/articles/10.1051/ps/2014008/} }
TY - JOUR AU - Marckert, Jean-François AU - Renault, David TI - Compact convex sets of the plane and probability theory JO - ESAIM: Probability and Statistics PY - 2014 SP - 854 EP - 880 VL - 18 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2014008/ DO - 10.1051/ps/2014008 LA - en ID - PS_2014__18__854_0 ER -
%0 Journal Article %A Marckert, Jean-François %A Renault, David %T Compact convex sets of the plane and probability theory %J ESAIM: Probability and Statistics %D 2014 %P 854-880 %V 18 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps/2014008/ %R 10.1051/ps/2014008 %G en %F PS_2014__18__854_0
Marckert, Jean-François; Renault, David. Compact convex sets of the plane and probability theory. ESAIM: Probability and Statistics, Tome 18 (2014), pp. 854-880. doi : 10.1051/ps/2014008. http://www.numdam.org/articles/10.1051/ps/2014008/
[1] Sylvester's question: The probability that n points are in convex position. Ann. Probab. 27 (1999) 2020-2034. | MR | Zbl
,[2] Random polytopes, convex bodies and approximation, in Stochastic Geometry, Vol. 1892 of Lect. Notes Math. Springer Berlin/Heidelberg (2007) 77-118. | MR | Zbl
,[3] On the number of convex lattice polytopes. Geom. Func. Anal. 2 (1992) 381-393. | MR | Zbl
and ,[4] 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). | MR | Zbl
,[5] 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
, , and ,[6] Universality of the limit shape of convex lattice polygonal lines. Ann. Probab. 39 (1992) 2271-2317. | MR | Zbl
and ,[7] 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. | MR | Zbl
,[8] Convex Surfaces. Interscience. New York (1958). | MR | Zbl
,[9] 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. | MR | Zbl
,[10] Real Analysis and Probability. Cambridge Studies in Advanced Mathematics. Cambridge University Press (2002). | MR | Zbl
,[11] An introduction to probability theory and its applications. Vol. II. 2nd edition. John Wiley & Sons Inc., New York (1971). | MR | Zbl
,[12] Sur le problème des isopérimètres. C. R. Acad. Sci. Paris 132 (1901) 401-403. | JFM
,[13] 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] On John-type ellipsoids, in Geometric aspects of functional analysis, vol. 1850 of Lect. Notes Math. Springer, Berlin (2004) 149-158. | MR | Zbl
,[15] 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. | MR | Zbl
,[16] 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. | JFM | Numdam | Zbl
,[17] Probability that n random points in a disk are in convex position. Available at http://arxiv.org/abs/1402.3512 (2014).
,[18] Selected Topics in Convex Geometry. Birkhäuser (2006). | Zbl
,[19] 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). | MR | Zbl
,[20] 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. | MR | Zbl
,[21] Isoperimetric Inequalities in Mathematical Physics. Ann. Math. Stud. Kraus (1965). | Zbl
,[22] Real and Complex Analysis, 3rd edn. McGraw-Hill International Editions (1987). | MR | Zbl
,[23] Convex Bodies: The Brunn−Minkowski Theory. Cambridge University Press (1993). | MR | Zbl
,[24] Probabilistic approach to the analysis of statistics for convex polygonal lines. Functional Anal. Appl. 28 (1994) 1. | MR | Zbl
,[25] On a special class of questions on the theory of probabilities. Birmingham British Assoc. Rept. (1865) 8-9.
,[26] Orthogonal polynomials. Colloquium Publications, 4th edition. American Mathematical Society (1939). | JFM
,[27] Probability that n random points are in convex position. Discr. Comput. Geom. 13 (1995) 637-643. | MR | Zbl
,[28] The probability that n random points in a triangle are in convex position. Combinatorica 16 (1996) 567-573. | MR | Zbl
,[29] A. Vershik and O. Zeitouni, large deviations in the geometry of convex lattice polygons. Israel J. Math. 109 (1999) 13-27. | MR | Zbl
[30] Fractional parts of random variables. Limit theorems and infinite divisibility, Dissertation. Technische Universiteit Eindhoven, Eindhoven (1994). | MR | Zbl
,Cité par Sources :