  During the last decade Optimal Transport had a relevant role in the study of
geometry of singular spaces that culminated with the Lott-Sturm-Villani theory.
The latter is built on the characterisation of Ricci curvature lower bounds in
terms of displacement convexity of certain entropy functionals along
$W_{2}$-geodesics. Substantial recent advancements in the theory (localization
paradigm and local-to-global property) have been obtained considering the
different point of view of $L^1$-Optimal transport problems yielding a
different curvature dimension $\mathsf{CD}^{1}(K,N)$ [8] formulated in terms of
one-dimensional curvature properties of integral curves of Lipschitz maps. In
this note we show that the two approaches produce the same curvature-dimension
condition reconciling the two definitions. In particular we show that the
$\mathsf{CD}^{1}(K,N)$ condition can be formulated in terms of displacement
convexity along $W_{1}$-geodesics.
