Gaussian measures of dilations of convex rotationally symmetric sets in $\mathbb{C}^n$

Abstract

We consider the complex case of the S-inequality. It concerns the behaviour of Gaussian measures of dilations of convex and rotationally symmetric sets in $\mathbb{C}^n$. We pose and discuss a conjecture that among all such sets measures of cylinders decrease the fastest under dilations. Our main result in this paper is that this conjecture holds under the additional assumption that the Gaussian measure of the sets considered is not greater than some constant $c > 0.64$.