# Kantorovich duality with pseudometrics

The usual framework for the Kantorovich duality in optimal transport theory uses Polish spaces as ground spaces for the distributions that should be transported. Are there results available that generalize the setting to pseudometric spaces? Or is this impossible? I will now specify the setting I am mainly interested in: For a Polish space $$X$$ and a metric $$d$$ on $$X$$ the cost $$T_d$$ of optimal transportation with (continuous) cost function $$c(x,y)=d(x,y)$$ is given by \begin{align*} T_d(\mu,\nu)=\inf_{\pi \in \Pi(\mu,\nu)} \int_{X \times X} d(x,y) d\pi(x,y). \end{align*}, where the $$\inf$$ is running over all joint distributions with marginals $$\mu$$ and $$\nu$$. Let $$Lip(X)$$ denote the set of all Lipschitz functions on $$X$$, and \begin{align} \| \varphi \|_{Lip}=\sup_{x\ne y} \frac{|\varphi(x)-\varphi(y)|}{d(x,y)}. \end{align} Then \begin{align} T_d(\mu,\nu)=\sup \{ \int_X \phi d(\mu-\nu) | \quad \phi \in L^1(d|\mu-\nu|), \| \varphi \|_{Lip}\le 1 \}. \end{align} My question now would be, if this conclusion also holds, if $$d$$ is a pseudometric instead of a metric. Mainly: Is it possible to draw immediate conclusions that follow from the proof in the metric case, or is this dependent on the topology induced by $$d$$

## 1 Answer

Yes the Kantorovich Duality holds for continuous cost functions by following the proof in Villani's book without any change. The proof for general cost functions needs compactness of the set of couplings between the marginals which follows from Prokhorov's Theorem but might be true in more general settings.

Being Polish (=complete separable metrizable space) is usual used for the existence of optimal couplings, e.g. Prokhorov's Theorem and the Portmanteau Lemma are the foundations. So you might want check whether the topological properties of your space allow you to obtain the existence of optimal couplings. Note a pseudometric space can be Polish as well.

• Thank you very much for the quick answer! For a more specific description of my problem, please consider the edited version of my original post! – Markus Holzleitner Dec 19 '18 at 14:43
• In this setting you may even assume the pseudometric is only lower semicontinuous as it is exactly the setting of Villani’s book and the majority of papers on optimal transport. A dual solution exists as well, but there might not be a dual solution if the cost function is infinite somewhere. – Martin Kell Dec 19 '18 at 14:51
• Also the application of Prokhorov's Theorem works without any problems in this setting? Villani considers Polish spaces in his book, but in my setting, as mentioned before, the ground space $X$ is now equipped with a pseudometric. In other words: as long as the cost function is lower semicontinuous, the generalization to pseudometric spaces instead of Polish spaces is straightforward? Did I understand this correctly? In any case, thanks a lot for helping me! – Markus Holzleitner Dec 19 '18 at 14:59
• Sorry by “this setting” I meant Polish spaces. There are metric spaces that are not Polish (any non-separable Hilbert/Banach space). There are even metric spaces that are incomplete but Polish (think of the sphere and remove a point, this is homeomophic to Euclidean space hence incomplete with the spherical metric but complete with the Euclidean one). – Martin Kell Dec 19 '18 at 15:09
• Note that the quotient space (spaces of equivalence classes induced by x~y iff d(x,y) is a metric space with metric induced by d. However it might be a very bad metric space, like one not satisfying Prokhorov. – Martin Kell Dec 19 '18 at 15:11