Compact convex sets of the plane and probability theory^∗

CNRS, LaBRI, Université de Bordeaux, 351 cours de la Libération, 33405 Talence cedex, France
name@labri.fr; renault@labri.fr

Received: 11 December 2012
Revised: 15 July 2013

Abstract

The Gauss−Minkowski correspondence in ℝ² states the existence of a homeomorphism between the probability measures μ on [0,2π] such that 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 ) 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.