On the Isoperimetric constant, covariance inequalities and L_p-Poincaré inequalities in dimension one
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Firstly, we derive in dimension one a new covariance inequality of L_1-L∞ 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 integer p.
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