Weighted Poincar\'{e}-type inequalities for Cauchy and other convex measures

TL;DR

This paper develops weighted Poincaré and isoperimetric inequalities for Cauchy and convex measures, revealing their concentration and deviation properties and extending classical results to a broader class of probability measures.

Contribution

It introduces new weighted inequalities for convex measures, including Cauchy distributions, and explores their implications for measure concentration and isoperimetric properties.

Findings

Weighted inequalities characterize measure concentration.

Cheeger-type inequalities are established for convex measures.

Results extend Gaussian inequalities to broader convex measures.

Abstract

Brascamp--Lieb-type, weighted Poincar\'{e}-type and related analytic inequalities are studied for multidimensional Cauchy distributions and more general $\kappa$-concave probability measures (in the hierarchy of convex measures). In analogy with the limiting (infinite-dimensional log-concave) Gaussian model, the weighted inequalities fully describe the measure concentration and large deviation properties of this family of measures. Cheeger-type isoperimetric inequalities are investigated similarly, giving rise to a common weight in the class of concave probability measures under consideration.