  In this essay, we discuss the notion of optimal transport on geodesic measure
spaces and the associated (2-)Wasserstein distance. We then examine
displacement convexity of the entropy functional on the space of probability
measures. In particular, we give a detailed proof that the Lott-Villani-Sturm
notion of generalized Ricci bounds agree with the classical notion on smooth
manifolds. We also give the proof that generalized Ricci bounds are preserved
under Gromov-Hausdorff convergence. In particular, we examine in detail the
space of probability measures over the interval, $P(X)$ equipped with the
Wasserstein metric $d^W$. We show that this metric space is isometric to a
totally convex subset of a Hilbert space, $L^2[0,1]$, which allows for concrete
calculations, contrary to the usual state of affairs in the theory of optimal
transport. We prove explicitly that $(P(X),d^W)$ has vanishing Alexandrov
curvature, and give an easy to work with expression for the entropy functional
on this space. In addition, we examine finite dimensional Gromov-Hausdorff
approximations to this space, and use these to construct a measure on the limit
space, the entropic measure first considered by Von Renesse and Sturm. We
examine properties of the measure, in particular explaining why one would
expect it to have generalized Ricci lower bounds. We then show that this is in
fact not true. We also discuss the possibility and consequences of finding a
different measure which does admit generalized Ricci lower bounds.
