%0 Journal Article %1 saumard2017isoperimetric %A Saumard, Adrien %A Wellner, Jon A. %D 2017 %K functional-inequalities %T On the isoperimetric constant, covariance inequalities and $L_p$-Poincaré inequalities in dimension one %U http://arxiv.org/abs/1711.00668 %X Firstly, we derive in dimension one a new covariance inequality of $L_1-L_ınfty$ 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 $L_p$-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 $p1$.

@article{saumard2017isoperimetric, 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 $p\geq 1$.}, added-at = {2018-03-09T15:19:07.000+0100}, author = {Saumard, Adrien and Wellner, Jon A.}, biburl = {https://www.bibsonomy.org/bibtex/252403c5ca22b4c1dee29be9d78091676/claired}, description = {On the isoperimetric constant, covariance inequalities and $L_p$-Poincar\'{e} inequalities in dimension one}, interhash = {e1f890e0cf548fd3e3f79edac77bee5a}, intrahash = {52403c5ca22b4c1dee29be9d78091676}, keywords = {functional-inequalities}, note = {cite arxiv:1711.00668}, timestamp = {2018-03-09T15:19:07.000+0100}, title = {On the isoperimetric constant, covariance inequalities and $L_p$-Poincar\'{e} inequalities in dimension one}, url = {http://arxiv.org/abs/1711.00668}, year = 2017 }