Strong convexity of internal energy with respect to Wasserstein metric

Question

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)?

Answer 1

This is a very delicate topics and requires a curvature condition on the underlying Polish space $X$ (upon which the probability space $\mathcal P(X)$ is based). This is known nowadays as the "Sturm-Lott-Villani" synthetic theory of curvature, see part III in Villani's (big) book. Long story short: requiring that $H(\mu)=\int_X\mu\log\mu\, \mathrm d vol$ be $\lambda$-convex is Sturm-Lott-Villani's generalized definition for the Ricci lower bound $R(X)\geq \lambda$.

For a more practical answer to your question: in the flat space $\mathbb R^d$ (or any reasonable subdomain thereof) the internal energy will in general not be $\lambda$-convex for ANY $\lambda>0$. In the sphere (with uniformly positive Ricci curvature), you can get a positive modulus of convexity.