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Normally distributed probability measure on the metric space of norms

In this paper we propose a method to construct probability measures on the space of convex bodies with a given pushforward distribution. Concretely we show that there is a measure on the metric space of centrally symmetric convex bodies, which pushforward by the thinness mapping produces a probability measure of truncated normal distribution on the interval of its range. Improving the construction we give another (more complicated) one with the following additional properties; the neighborhoods have positive measure, the set of polytopes has zero measure and the set of smooth bodies has measure 1, respectively.