Compact convex sets of the plane and probability theory

Abstract

The Gauss-Minkowski correspondence in R 2 states the existence of a homeomorphism between the probability measures µ on [0, 2π] such that 2π 0 e ix dµ(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 2π 0 e ix dµ(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.