Sectional curvature and matrix displacement convexity

We will discuss a recent paper of Aishwarya, Rotem, and Shenfeld, in which they introduce a notion of Matrix displacement convexity of entropy on the Wasserstein space of a Riemannian manifold. This property is shown to be equivalent to nonnegative sectional curvature of the manifold. This fact is then used to prove some intrinsic-dimensional geometric inequalities on nonnegatively curved Riemannian manifolds.