Isoperimetric profile of radial probability measures on Euclidean spaces

TL;DR

This paper derives the isoperimetric profile for radial probability measures in Euclidean spaces, generalizing the Poincaré limit to include certain maps beyond projections, with implications for Gaussian-type measures.

Contribution

It introduces a generalized approach to the Poincaré limit, extending the understanding of isoperimetric profiles for radial measures beyond Gaussian cases.

Findings

Derived the isoperimetric profile for Gaussian-type radial measures.

Generalized the Poincaré limit using specific maps.

Provided new tools for analyzing measures on Euclidean spaces.

Abstract

We derive the isoperimetric profile of Gaussian type for an absolutely continuous probability measure on Euclidean spaces with respect to the Lebesgue measure, whose density is a radial function.The key is a generalization of the Poincar\'e limit which asserts that the $n$-dimensional Gaussian measure is approximated by the projections of the uniform probability measure on the Euclidean sphere of appropriate radius to the first $n$-coordinates as the dimension diverges to infinity. The generalization is done by replacing the projections with certain maps.