It is well known that the internal energy (see, e.g., Definition 3.32 in and Proposition 3.33 in 1) is geodesically convex with the 2-Wasserstein distance. I was wondering under what condition, the internal energy functional is $\lambda$-convex where $\lambda>0$ (see the top of Page 55 in 1 for the definition of a $\lambda$-convex functional)?