Normally distributed probability measure on the metric space of norms

TL;DR

This paper introduces methods to construct probability measures on the space of convex bodies, specifically producing a truncated normal distribution via the thinness mapping, with refined properties for neighborhoods, polytopes, and smooth bodies.

Contribution

It presents novel constructions of probability measures on convex bodies that produce a normal distribution under a specific mapping, with improved measure-theoretic properties.

Findings

Existence of a measure on centrally symmetric convex bodies with a truncated normal pushforward

A refined construction ensuring neighborhoods have positive measure

The set of polytopes has zero measure, and smooth bodies have measure one

Abstract

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.