Isoperimetric inequality for radial probability measures on Euclidean spaces

Abstract

We generalize the Poincaré 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. Using this generalization, we derive a Gaussian isoperimetric inequality for an absolutely continuous probability measure on Euclidean spaces with respect to the Lebesgue measure, whose density is a radial function.