Isoperimetric profile of radial probability 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.