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