On the isoperimetric constant, covariance inequalities and Lp-Poincare inequalities in dimension one

First, we derive in dimension one a new covariance inequality of L-1 - L-infinity 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-Poincare inequalities and moment bounds. In particular, we obtain optimal constants in general L-p-Poincare inequalities for measures with finite isoperimetric constant, thus generalizing in dimension one Cheeger's inequality, which is a L-p-Poincare inequality for p = 2, to any real p >= 1.