NORMALLY DISTRIBUTED PROBABILITY MEASURE ON THE METRIC SPACE OF NORMS∗

Abstract In this paper we propose a method to construct probability measures on the space of convex bodies. For this purpose, first, we introduce the notion of thinness of a body. Then we show the existence of a measure with the property that its pushforward by the thinness function is a probability measure of truncated normal distribution. Finally, we improve this method to find a measure satisfying some important properties in geometric measure theory. Key words Hausdorff metric; Borel; Dirac; Haar and Lebesgue-measure; space of convex bodies; metric space of norm