  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
$p\geq 1$.
