Gaussian measures of dilations of convex rotationally symmetric sets in \(C^n\). (English) Zbl 1225.60059

Tkocz, Tomasz

Electron. Commun. Probab. 16, 38-49 (2011).

Summary: 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 the conjecture that among all such sets, the measures of cylinders decrease the fastest under dilations. Our main result is that this conjecture holds under the additional assumption that the Gaussian measure of the sets considered is not greater than some constant \(c>0.64\).

MSC:

60G15	Gaussian processes
60E15	Inequalities; stochastic orderings

Keywords:

Gaussian measure; convex bodies; isoperimetric inequalities