## Bernoulli

- Bernoulli
- Volume 25, Number 3 (2019), 1794-1815.

### On the isoperimetric constant, covariance inequalities and $L_{p}$-Poincaré inequalities in dimension one

Adrien Saumard and Jon A. Wellner

#### Abstract

First, 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. Second, 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 $L_{p}$-Poincaré inequalities and moment bounds. In particular, we obtain optimal constants in general $L_{p}$-Poincaré inequalities for measures with finite isoperimetric constant, thus generalizing in dimension one Cheeger’s inequality, which is a $L_{p}$-Poincaré inequality for $p=2$, to any real $p\geq1$.

#### Article information

**Source**

Bernoulli, Volume 25, Number 3 (2019), 1794-1815.

**Dates**

Received: November 2017

Revised: March 2018

First available in Project Euclid: 12 June 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1560326428

**Digital Object Identifier**

doi:10.3150/18-BEJ1036

**Mathematical Reviews number (MathSciNet)**

MR3961231

**Zentralblatt MATH identifier**

07066240

**Keywords**

Cheeger’s inequality covariance formula covariance inequality isoperimetric constant moment bounds Poincaré inequality

#### Citation

Saumard, Adrien; Wellner, Jon A. On the isoperimetric constant, covariance inequalities and $L_{p}$-Poincaré inequalities in dimension one. Bernoulli 25 (2019), no. 3, 1794--1815. doi:10.3150/18-BEJ1036. https://projecteuclid.org/euclid.bj/1560326428