Logarithmic Sobolev inequalities for generalised Cauchy measures

TL;DR

This paper establishes logarithmic Sobolev inequalities for generalized Cauchy measures using a curvature-dimension criterion, providing explicit constants and deriving concentration results, including the optimal one-dimensional case.

Contribution

It introduces a curvature-dimension criterion for generalized Cauchy measures and derives explicit, optimal constants for the inequalities.

Findings

Logarithmic Sobolev inequalities with explicit constants for generalized Cauchy measures.

Optimal constant achieved in the one-dimensional case.

Concentration results derived from these inequalities.

Abstract

We prove a curvature-dimension criterion and obtain logarithmic Sobolev inequalities for generalised Cauchy measures with optimal weights and explicit constants. In the one-dimensional case, this constant is even optimal. From these inequalities, we derive concentration results, which allow concluding the case of the pathological dimension two.