# Strict convexity of the cost function is enough to ensure the existence and uniqueness of the optimal transport map

Let $$X=Y = \mathbb R^d$$ and $$c:X \times Y \to [0, \infty)$$ be Borel measurable. Let $$\mu, \nu$$ be Borel probability measures on $$X,Y$$ respectively. Let $$\mathcal T$$ be the set of all Borel measurable maps $$T:X \to Y$$ such that $$T_\sharp \mu = \nu$$. Let $$\mathbb M (T) := \int_X c(x, T(x)) \mathrm d \mu(x) \quad \forall T \in \mathcal T.$$

Then we are interested in the Monge's transportation problem $$\mathrm{MP} : \quad\inf_{T \in \mathcal T} \mathbb M (T).$$

The existence and uniqueness of the solution of $$\mathrm{MP}$$ is guaranteed if $$c$$ is strictly convex and the supports of $$\mu, \nu$$ are compact . We can remove the assumption of compact supports by stronger conditions on $$c$$ .

Let $$\Pi (\mu, \nu)$$ be the set of Borel probability measures on $$X \times Y$$ with marginals $$\mu$$ on $$X$$ and $$\nu$$ on $$Y$$. Recently, I have seen a strong theorem from this lecture note, i.e.,

Theorem 3.14. Assume

1. $$h:\mathbb R^d \to [0, \infty)$$ is strictly convex and $$c(x, y) := h(x-y)$$ for all $$(x, y) \in X \times Y$$.
2. $$\int_{X \times Y} c \mathrm{d} \gamma <\infty$$ for some $$\gamma \in \Pi(\mu, \nu)$$,
3. $$\mu\left(\left\{x \in X: \int_Y c(x, y) \mathrm{d} \nu(y)<\infty\right\}\right)>0$$,
4. $$\nu\left(\left\{y \in Y: \int_X c(x, y) \mathrm{d} \mu(x)<\infty\right\}\right)>0$$,
5. $$\mu$$ is absolutely continuous with respect to Lebesgue measure.

Then $$\mathrm{MP}$$ has a unique (up to $$\mu$$-a.e. solution).

The conditions 2, 3, 4 are very mild and just to ensure the dual of the dual of the corresponding Kantorovich's problem has a solution in a form of a pair of $$c$$-conjugates. Theorem 3.14. is striking because it does not require the supports of $$\mu, \nu$$ to be compact nor any condition on $$c$$ besides strict convexity.

Could you elaborate if there are some references of Theorem 3.14.?

 Caffarelli, Luis A. "Allocation maps with general cost functions." Partial differential equations and applications. Routledge, 2017. 29-35.

 Gangbo, Wilfrid, and Robert J. McCann. "The geometry of optimal transportation." Acta Mathematica 177.2 (1996): 113-161.

First a comment: you write (before stating your Theorem 3.14) that "The existence and uniqueness of the solution of the Monge Problem is guaranteed if $$c$$ is strictly convex and the supports of $$\mu,\nu$$ are compact", but this is absolutely not true: you really need some conditions on the starting point, i-e that $$\mu$$ does not charge "small sets" in some sense (this is exactly assumption 5 in your theorem 3.14, but this can be relaxed). If $$\mu=\delta_x$$ is a Dirac delta there exists no transport map $$T$$ from $$\mu$$ to $$\nu$$, unless $$\nu=\delta_{T(x)}$$, so clearly the Monge problem is ill-posed in general.
Next, my real answer: this precise statement is often called the Brenier-McCann theorem. You can find an extremely general version in , Theorem 10.38. Note in particular that this is stated without any compactness assumptions or behaviour at infinity, and that the strict convexity $$c(x,y)=h(|x-y|)$$ is not needed (only the so-called and weaker twist condition). This result is usually credited to Brenier, Rachev and Rüschendorf for the quadratic cost in Euclidean spaces, and then R. McCann extended to Riemannian manifolds.
• It seems the assumption (Super) ($c$ is superdifferentiable everywhere) may not be salified for the cost function $c(x, y) := h(x-y)$ with $h$ being strictly convex... Jan 9 at 17:25
• Agreed, but it is in practice satisfied for all the reasonably smooth cases, e.g. as soon as $h$ is actually differentiable. Are you interested in a borderline case where this fails? Jan 9 at 17:43
• I just would like to verify Theorem 3.14. in the lecture note. Of course, convex function on $\mathbb R^n$ is locally Lipschitz and thus differentiable almost everywhere w.r.t. Lebesgue measure $\lambda$. I wonder if the condition (Super) can be weakened to "$c$ is superdifferentiable $\lambda$-a.e." If not, then I think Theorem 3.14. is not true as stated. Jan 9 at 17:54