Inner Regularization of Log-Concave Measures and Small-Ball Estimates

TL;DR

This paper introduces an inner regularization technique for log-concave measures using convolution with a random orthogonal image, preserving small-ball properties while achieving super-Gaussian marginals, and applies it to recover small-ball estimates.

Contribution

It proposes a novel inner-thickening method for log-concave measures that maintains small-ball information, unlike traditional Gaussian convolution.

Findings

Recovered Paouris' small-ball estimates

Preserved small-ball properties with inner-thickening

Provided a new preprocessing step for measure regularization

Abstract

In the study of concentration properties of isotropic log-concave measures, it is often useful to first ensure that the measure has super-Gaussian marginals. To this end, a standard preprocessing step is to convolve with a Gaussian measure, but this has the disadvantage of destroying small-ball information. We propose an alternative preprocessing step for making the measure seem super-Gaussian, at least up to reasonably high moments, which does not suffer from this caveat: namely, convolving the measure with a random orthogonal image of itself. As an application of this "inner-thickening", we recover Paouris' small-ball estimates.