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.