  In this article, a proof of the interpolation inequality along geodesics in
$p$-Wasserstein spaces is given. This interpolation inequality was the main
ingredient to prove the Borel-Brascamp-Lieb inequality for general Riemannian
and Finsler manifolds and led Lott-Villani and Sturm to define an abstract
Ricci curvature condition. Following their ideas, a similar condition can be
defined and for positively curved spaces one can prove a Poincar\'e inequality.
Using Gigli's recently developed calculus on metric measure spaces, even a
$q$-Laplacian comparison theorem holds on $q$-infinitesimal convex spaces. In
the appendix, the theory of Orlicz-Wasserstein spaces is developed and
necessary adjustments to prove the interpolation inequality along geodesics in
those spaces are given.
