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

Firstly, we derive in dimension one a new covariance inequality of $L_1-L_{\mathrm{\infty }}$ 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 $p\ge 1$.