Let $W(\mu,\nu)^2_2$ be the 2-wasserstein distance between probability distributions $\mu,\nu$ over $\mathbb{R}^d$ such that $d\mu,d\nu\ll\text{dVol}$:

$$ W(\mu,\nu)^2_2=\inf_{\pi\in\Pi(\mu,\nu)} \iint \|x-y\|^2 d\pi(x,y).$$

Let $MP(\mu,\nu)=\{T: \mathbb{R}^d\longrightarrow \mathbb{R}^d| T\#\mu=\nu \}.$

Brenier's theorem states that:

There exists a unique (up to a $\mu$-negligible set) minimizer $T^*$ to the problem $$d(\mu,\nu)^2=\inf_{T\in MP(\mu,\nu)} \int \|x-T(x)\|^2 d\mu(x)$$ such that $d(\mu,\nu)^2=W(\mu,\nu)^2_2$, and $T^*$ can be represented $\mu$-almost everywhere as $T^*=\nabla \psi$ for some convex function $\psi:\mathbb{R}^d\longrightarrow \mathbb{R}$.


Imagine that one introduces a Riemannian metric $g$ in $\mathbb{R}^d$, so that all Euclidean distances appearing above are now replaced by geodesic distances w.r.t. $g$ . Suppose that $\mu,\nu$ are two probability measures on $(\mathbb{R}^d,g)$.

Is there an analogous statement to the one above? Namely that there exists a unique $T^*$ that is the gradient of a convex function, and that $d(\mu,\nu)^2$ and $W(\mu,\nu)^2_2$ coincide?

  • 1
    $\begingroup$ I think you will have a better chance of receiving an answer by moving your question to mathoverflow.net. $\endgroup$
    – Akira
    Commented Jun 8, 2023 at 7:10
  • 1
    $\begingroup$ There is an appropriate generalisation of Brenier's theorem in Villani's OT book. $\endgroup$
    – Tobsn
    Commented Jun 11, 2023 at 14:28


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