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The Gauss−Minkowski correspondence in ℝ states the existence of a homeomorphism between the probability measures on [0,2] such that ${\int }_0^{2\pi }{\mathrm{e}}^{ix}\mathrm{d}\mu \left(x\right)=0$ ∫ 0 2 π e ix d μ ( 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...