A Comparative Analysis on Gaussian Isoperimetric Inequality, And Its Similar Concentration Trend: Exploring Properties and Applications of Gaussian Measures

Abstract

The Gaussian isoperimetric inequality, and its related concentration phenomenon, is one of the most important properties of Gaussian measures. These notes aim to present, in a concise and self-contained form, the fundamental results on Gaussian processes and measures based on the isoperimetric tool. In particular, our exposition will include, from this modern point of view, some of the by now classical aspects such as inerrability and tail behavior of Gaussian semi norms, large deviations or regularity of Gaussian sample paths. We will also concentrate on some of the more recent aspects of the theory which deal with small ball probabilities.