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.