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Compact convex sets of the plane and probability theory∗

Published online by Cambridge University Press:  29 October 2014

Jean-François Marckert  and
David Renault
Jean-François Marckert
Affiliation:
CNRS, LaBRI, Université de Bordeaux, 351 cours de la Libération, 33405 Talence cedex, France. name@labri.fr; renault@labri.fr
David Renault
Affiliation:
CNRS, LaBRI, Université de Bordeaux, 351 cours de la Libération, 33405 Talence cedex, France. name@labri.fr; renault@labri.fr
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Abstract

The Gauss−Minkowski correspondence in ℝ2 states the existence of a homeomorphism between the probability measures μ on [0,2π] such that \hbox{$\int_0^{2\pi} {\rm e}^{ix}{\rm d}\mu(x)=0$}∫02πeixdμ(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 \hbox{$\int_0^{2\pi} {\rm e}^{ix}{\rm d}\mu(x)=0$}∫02πeixdμ(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.

Keywords

Random convex setssymmetrisationweak convergenceMinkowski sum
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 18 , 2014 , pp. 854 - 880
Copyright
© EDP Sciences, SMAI 2014

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