• 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

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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

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

Received: November 2017
Revised: March 2018
First available in Project Euclid: 12 June 2019

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


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.

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