  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.
