  We prove that for a probability measure on $\mathbb{R}^n$, the Poincar\'e
inequality for convex functions is equivalent to the weak transportation
inequality with a quadratic-linear cost. This generalizes recent results by
Gozlan et al. and Feldheim et al., concerning probability measures on the real
line. The proof relies on modified logarithmic Sobolev inequalities of
Bobkov-Ledoux type for convex and concave functions, which are of independent
interest. We also present refined concentration inequalities for general (not
necessarily Lipschitz) convex functions, complementing recent results by
Bobkov, Nayar and Tetali.
