Open Access
August 2019 On the isoperimetric constant, covariance inequalities and $L_{p}$-Poincaré inequalities in dimension one
Adrien Saumard, Jon A. Wellner
Bernoulli 25(3): 1794-1815 (August 2019). DOI: 10.3150/18-BEJ1036


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 $p\geq1$.


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Adrien Saumard. Jon A. Wellner. "On the isoperimetric constant, covariance inequalities and $L_{p}$-Poincaré inequalities in dimension one." Bernoulli 25 (3) 1794 - 1815, August 2019.


Received: 1 November 2017; Revised: 1 March 2018; Published: August 2019
First available in Project Euclid: 12 June 2019

zbMATH: 07066240
MathSciNet: MR3961231
Digital Object Identifier: 10.3150/18-BEJ1036

Keywords: Cheeger’s inequality , covariance formula , covariance inequality , isoperimetric constant , moment bounds , Poincaré inequality

Rights: Copyright © 2019 Bernoulli Society for Mathematical Statistics and Probability

Vol.25 • No. 3 • August 2019
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