# Compact convex sets of the plane and probability theory

Jean-François Marckert; David Renault

ESAIM: Probability and Statistics (2014)

- Volume: 18, page 854-880
- ISSN: 1292-8100

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topMarckert, Jean-François, and Renault, David. "Compact convex sets of the plane and probability theory." ESAIM: Probability and Statistics 18 (2014): 854-880. <http://eudml.org/doc/274360>.

@article{Marckert2014,

abstract = {The Gauss−Minkowski correspondence in ℝ2 states the existence of a homeomorphism between the probability measures μ on [0,2π] such that $\int _0^\{2\pi \} \{\rm e\}^\{ix\}\{\rm d\}\mu (x)=0$ ∫ 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 ofnrandom variables (satisfying $\int _0^\{2\pi \} \{\rm e\}^\{ix\}\{\rm d\}\mu (x)=0$ ∫ 0 2 π e ix d μ ( x ) = 0 ) converges to a CCS associated withμ 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.},

author = {Marckert, Jean-François, Renault, David},

journal = {ESAIM: Probability and Statistics},

keywords = {random convex sets; symmetrisation; weak convergence; Minkowski sum; probability measure},

language = {eng},

pages = {854-880},

publisher = {EDP-Sciences},

title = {Compact convex sets of the plane and probability theory},

url = {http://eudml.org/doc/274360},

volume = {18},

year = {2014},

}

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

PB - EDP-Sciences

VL - 18

SP - 854

EP - 880

AB - The Gauss−Minkowski correspondence in ℝ2 states the existence of a homeomorphism between the probability measures μ on [0,2π] such that $\int _0^{2\pi } {\rm e}^{ix}{\rm d}\mu (x)=0$ ∫ 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 ofnrandom variables (satisfying $\int _0^{2\pi } {\rm e}^{ix}{\rm d}\mu (x)=0$ ∫ 0 2 π e ix d μ ( x ) = 0 ) converges to a CCS associated withμ 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.

LA - eng

KW - random convex sets; symmetrisation; weak convergence; Minkowski sum; probability measure

UR - http://eudml.org/doc/274360

ER -

## References

top- [1] I. Bárány, Sylvester’s question: The probability that n points are in convex position. Ann. Probab.27 (1999) 2020–2034. Zbl0959.60006MR1742899
- [2] I. Bárány, Random polytopes, convex bodies and approximation, in Stochastic Geometry, Vol. 1892 of Lect. Notes Math. Springer Berlin/Heidelberg (2007) 77–118. Zbl1123.60006MR2327291
- [3] I. Bárány and A.M. Vershik, On the number of convex lattice polytopes. Geom. Func. Anal.2 (1992) 381–393. Zbl0772.52010MR1191566
- [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). Zbl0172.21201MR1700749
- [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. Zbl06169451MR3155272
- [6] L.V. Bogachev and S.M. Zarbaliev, Universality of the limit shape of convex lattice polygonal lines. Ann. Probab.39 (1992) 2271–2317. Zbl1242.52007MR2932669
- [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. Zbl1247.52005MR2911166
- [8] H. Busemann, Convex Surfaces. Interscience. New York (1958). Zbl0196.55101MR105155
- [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. Zbl1045.60005MR1990603
- [10] R.M. Dudley, Real Analysis and Probability. Cambridge Studies in Advanced Mathematics. Cambridge University Press (2002). Zbl1023.60001MR1932358
- [11] W. Feller, An introduction to probability theory and its applications. Vol. II. 2nd edition. John Wiley & Sons Inc., New York (1971). Zbl0138.10207MR270403
- [12] M.A. Hurwitz, Sur le problème des isopérimètres. C. R. Acad. Sci. Paris132 (1901) 401–403. JFM32.0386.01
- [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. Zbl33.0599.02JFM33.0599.02
- [14] B. Klartag, On John-type ellipsoids, in Geometric aspects of functional analysis, vol. 1850 of Lect. Notes Math. Springer, Berlin (2004) 149–158. Zbl1067.52004MR2087157
- [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. Zbl0777.68012MR1226891
- [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. Zbl0023.05801JFM65.1346.01
- [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). Zbl1093.52001
- [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). Zbl0322.60042MR388499
- [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. Zbl0311.53067MR346714
- [21] G. Pólya, Isoperimetric Inequalities in Mathematical Physics. Ann. Math. Stud. Kraus (1965). Zbl0044.38301
- [22] W. Rudin, Real and Complex Analysis, 3rd edn. McGraw-Hill International Editions (1987). Zbl0278.26001MR924157
- [23] R. Schneider, Convex Bodies: The Brunn−Minkowski Theory. Cambridge University Press (1993). Zbl1287.52001MR1216521
- [24] Ya.G. Sinai, Probabilistic approach to the analysis of statistics for convex polygonal lines. Functional Anal. Appl. 28 (1994) 1. Zbl0832.60099MR1283251
- [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). JFM65.0278.03
- [27] P. Valtr, Probability that n random points are in convex position. Discr. Comput. Geom.13 (1995) 637–643. Zbl0820.60007MR1318803
- [28] P. Valtr, The probability that n random points in a triangle are in convex position. Combinatorica16 (1996) 567–573. Zbl0881.60010MR1433643
- [29] A. Vershik and O. Zeitouni, large deviations in the geometry of convex lattice polygons. Israel J. Math. 109 (1999) 13–27. Zbl0945.60022MR1679585
- [30] R.J.G. Wilms, Fractional parts of random variables. Limit theorems and infinite divisibility, Dissertation. Technische Universiteit Eindhoven, Eindhoven (1994). Zbl0804.60024MR1296668

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