On the isoperimetric constant, covariance inequalities and $L_p$-Poincar\'{e} inequalities in dimension one

TL;DR

This paper introduces new covariance inequalities in dimension one that characterize the isoperimetric constant, generalize Cheeger's inequality, and lead to optimal $L_p$-Poincaré inequalities for measures with finite isoperimetric constant.

Contribution

The paper derives novel $L_{p}-L_{q}$ covariance bounds, generalizes Cheeger's inequality to all $p \, \geq 1$, and establishes optimal constants in $L_p$-Poincaré inequalities.

Findings

New covariance inequalities characterizing the isoperimetric constant.

Generalization of Cheeger's inequality to all $p \geq 1$.

Optimal constants in $L_p$-Poincaré inequalities for measures with finite isoperimetric constant.

Abstract

Firstly, we derive in dimension one a new covariance inequality of $L_{1}-L_{\infty}$ type that characterizes the isoperimetric constant as the best constant achieving the inequality. Secondly, we generalize our result to $L_{p}-L_{q}$ bounds for the covariance. Consequently, we recover Cheeger's inequality without using the co-area formula. We also prove a generalized weighted Hardy type inequality that is needed to derive our covariance inequalities and that is of independent interest. Finally, we explore some consequences of our covariance inequalities for $L_{p}$-Poincar\'{e} inequalities and moment bounds. In particular, we obtain optimal constants in general $L_{p}$-Poincar\'{e} inequalities for measures with finite isoperimetric constant, thus generalizing in dimension one Cheeger's inequality, which is a $L_{p}$-Poincar\'{e} inequality for $p=2$, to any real $p\geq 1$.