Regularization inequalites for one-dimensional Cauchy-type measures

TL;DR

This paper explores inequalities related to one-dimensional Cauchy measures and their sections, extending previous work on boundary measures and drawing analogies with Gaussian measure inequalities.

Contribution

It introduces new inequalities for one-dimensional Cauchy measures and their sections, extending Gaussian measure results to the Cauchy context.

Findings

Identified extremal boundary measures for fixed measure intervals

Established inequalities analogous to Gaussian measure inequalities

Extended results to multidimensional isotropic Cauchy measures

Abstract

In the paper we investigate various inequalities for the one-dimensional Cauchy measure. We also consider analogous properties for one-dimensional sections of multidimensional isotropic Cauchy measure. The paper is a continuation of our previous investigations \cite{BZ}, where we found, among intervals with fixed measure, the ones with the extremal measure of the boundary. Here for the above mentioned measures we investigate inequalities that are analogous to those found for Gaussian measures by Borell in \cite{B} and by Landau and Shepp in \cite{LS}.