Equivalence of the Poincaré inequality with a transport-chi-square inequality in dimension one

In this paper, we prove that, in dimension one, the Poincaré inequality is equivalent to a new transport-chi-square inequality linking the square of the quadratic Wasserstein distance with the chi-square pseudo-distance. We also check tensorization of this transport-chi-square inequality. For q ≥ 1, the Wasserstein distance with index q between two probability measures µ and ν on R d is denoted by W q q (µ, ν) = inf γ< µ ν R d ×R d |x − y| q dγ(x, y) (0.1) where the infimum is taken over all probability measures γ on R d × R d with respective marginals µ and ν. We also introduce the relative entropy and the chi-square pseudo distance H(ν|µ) = R d ln dν dµ (x) dν(x) if ν absolutely continuous w.r.t. µ +∞ otherwise χ 2 2 (ν|µ) =    R d dν dµ (x) − 1 2 dµ(x) = dν dµ − 1 2 L 2 (µ) if ν absolutely continuous w.r.t. µ +∞ otherwise. Next, we precise the inequalities that will be discussed in the paper.