The Poincar\'e inequality and quadratic transportation-variance inequalities

It is known that the Poincar\'e inequality is equivalent to the quadratic transportation-variance inequality (namely $W_2^2(f\mu,\mu) \leqslant C_V \mathrm{Var}_\mu(f)$), see Jourdain \cite{Jourdain} and most recently Ledoux \cite{Ledoux18}. We give two alternative proofs to this fact. In particular, we achieve a smaller $C_V$ than before, which equals the double of Poincar\'e constant. Applying the same argument leads to more characterizations of the Poincar\'e inequality. Our method also yields a by-product as the equivalence between the logarithmic Sobolev inequality and strict contraction of heat flow in Wasserstein space provided that the Bakry-\'Emery curvature has a lower bound (here the control constants may depend on the curvature bound). Next, we present a comparison inequality between $W_2^2(f\mu,\mu)$ and its centralization $W_2^2(f_c\mu,\mu)$ for $f_c = \frac{|\sqrt{f} - \mu(\sqrt{f})|^2}{\mathrm{Var}_\mu (\sqrt{f})}$, which may be viewed as some special counterpart of the Rothaus' lemma for relative entropy. Then it yields some new bound of $W_2^2(f\mu,\mu)$ associated to the variance of $\sqrt{f}$ rather than $f$. As a by-product, we have another proof to derive the quadratic transportation-information inequality from Lyapunov condition, avoiding the Bobkov-G\"otze's characterization of the Talagrand's inequality.