# Properties of the dual optimizer in Wasserstein loss

Let $$P=\frac 1 k \sum_i N(\theta_i, \sigma^2 I)$$ and $$Q=\frac 1 k \sum_i N(\mu_i, \sigma^2 I)$$ be two mixtures of Gaussians in $$\mathbb{R}^d$$. A well known fact is that the Wasserstein-$$2$$ loss between $$P$$ and $$Q$$ can be written using the dual formulation as $$W_2^2(P,Q)=\sup_{(f,g):f(x)+g(y) \leq \|x-y\|^2} \int f(x) P(x) dx+\int g(y) Q(y) dy\\ =\int f(x) P(x) dx+\int f^c(y) Q(y) dy,$$ where $$f^c$$ denotes the $$c$$-transform of the function $$f$$. It is also known that the optimal $$f$$ is related to the optimal Monge transportation map $$T$$ and the Kantorovich convex potential $$u$$ via $$T(x) = x- \frac{\nabla f(x)}{2}, \quad T(x) = \nabla u(x) \Rightarrow \nabla^2 f(x) \preccurlyeq 2 I,$$ since $$\nabla^2 u(x) \succcurlyeq 0$$ from the convexity of $$u$$.

My question is regarding if any other gradient and Hessian information about the function $$f$$ is known in the literature. In particular I am looking for bounds of the type $$\nabla^2 f(x) \succcurlyeq -\alpha I$$ for some $$\alpha \geq 0$$. This is equivalent to a smoothness condition on the convex potential $$u$$, i.e $$u$$ is $$\alpha$$ smooth. Please point me to the relevant references if you are aware of any.