Gaussian measures of dilations of convex rotationally symmetric sets in C^n

TL;DR

This paper investigates the behavior of Gaussian measures of dilated convex, rotationally symmetric sets in complex n-space, proposing a conjecture about cylinders and proving it under a specific measure threshold.

Contribution

It introduces a conjecture on the measure decay of cylinders among symmetric convex sets and proves it under a measure constraint.

Findings

Conjecture holds for measures not exceeding 0.64.

Cylinders decrease fastest under dilation among symmetric convex sets.

Results extend understanding of Gaussian measures in complex spaces.

Abstract

We consider the complex case of the so-called S-inequality. It concerns the behaviour of the Gaussian measures of dilations of convex and rotationally symmetric sets in C^n (rotational symmetry is invariance under the multiplication by $e^{it}$, for any real t). We pose and discuss a conjecture that among all such sets the measure of cylinders (i.e. the sets $\{z \in C^n: |z_1| \leq p\}$) decrease the fastest under dilations. Our main result of the paper is that this conjecture holds under the additional assumption that the Gaussian measure of considered sets is not greater than some constant c > 0.64.