Compact convex sets of the plane and probability theory∗

Author： Marckert Jean-François Renault David

E-ISSN： 1262-3318|18|issue|854-880

ISSN： 1292-8100

Source： ESAIM: Probability and Statistics, Vol.18, Iss.issue, 2014-10, pp. : 854-880

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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{$\int_0^{2\pi} {\rm e}^{ix}{\rm d}\mu(x)=0$} 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{$\int_0^{2\pi} {\rm e}^{ix}{\rm d}\mu(x)=0$}) 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.