# Uniqueness of Kantorovich potentials?

$$\newcommand{\R}{\mathbb R}$$Take $$\Omega\subset \R^d$$ bounded, convex, and smooth. Consider the optimal transport problem with cost $$c(x,y)=\lvert x-y\rvert^2$$, leading to the quadratic Wassersein distance. Are there known optimal (?) conditions on the probability measures $$\mu,\nu\in \mathcal P(\Omega)$$ guaranteeing uniqueness (up to additive constants) of the Kantorovich potentials $$\varphi$$ from $$\mu$$ to $$\nu$$?

One standard answer is to require that $$\mu$$ be absolutely continuous w.r.t the $$d$$-dimensional Lebesgue measure $$\mathcal L^d_{\Omega}$$. Indeed a Kantorovich potential $$\varphi$$ is always Lipschitz (at least in bounded domains) hence differentiable Lebesgue-almost everywhere (Rademacher's theorem), and therefore also $$\mu$$-almost everywhere. Then one can prove that $$y-x=\nabla\varphi(x)$$ for any $$(x,y)$$ in the support of any optimal plan, which gives uniqueness of both the plans and potentials (and as a byproduct that $$T(x)=x+\nabla\varphi(x)$$ is the optimal Monge's map s.t. $$T_\#\mu=\nu$$). This is well explained e.g. in Filippo Santambrogio's book [1, theorem 1.17 pp. 15].

However, for my specific purpose I'm interested in a case where $$\mu=f_\mu \mathcal L^d\rvert_\Omega+g_\mu\mathcal L^{d-1}\rvert_{\partial\Omega}$$ and $$\nu=f_\nu \mathcal L^d\rvert_\Omega+g_\nu\mathcal L^{d-1}\rvert_{\partial\Omega}$$ typically have a nice smooth absolutely continuous part in the interior $$\Omega$$ but also a $$(d-1)$$-dimensional component on the boundary ($$\mathcal L^k$$ denotes here the $$k$$-dimensional Lebesgue measure). The above sufficient condition obviously fails, and I am left wondering what can be said? I still expect somehow that the Kantorovich potential (which always exists) is unique, but I have no real clue how to proceed and I was hoping someone out there would have a "black-box reference" that I could use before reinventing the wheel.

 Santambrogio, Filippo, Optimal transport for applied mathematicians. Calculus of variations, PDEs, and modeling, Progress in Nonlinear Differential Equations and Their Applications 87. Cham: Birkhäuser/Springer (ISBN 978-3-319-20827-5/hbk; 978-3-319-20828-2/ebook). xxvii, 353 p. (2015). ZBL1401.49002.

• You have uniqueness of the Monge OT plan $\pi$, and in your setting the uniqueness of the subdifferentials of the dual Kantorovich potentials $\phi, \psi$. So why trouble yourself with uniqueness of the potentials themselves. Honestly why? It's very complicated question and IMO a waste of time. Are you just curious, or looking for research project, or do you have an actual application in mind? I would politely ask Jun Kitagawa or Robert McCann or Brendan Pass, if they know the answer. I've never seen them on MO.
– JHM
Jun 10, 2022 at 14:11
• @JHM: I have my own application in mind, thank you very much. I just think it is not really worth explaining here (some nonstandard parabolic problem, formally a Wasserstein gradient flow). For the technical aspect: I need to take a first variation (Fréchet derivative) $\frac{\delta W^2(\mu,\nu)}{\delta\mu}$ with respect to $\mu$ ($\nu$ being fixed). IF the Kantorovich potential is unique then this first variation is indeed given by (the flat, $L^2$ action of) $\varphi_\mu$. Before bothering someone in person I though it was worth trying here on math.MO Jun 12, 2022 at 5:04
• MathJax note: a $\newcommand\…{…}$ followed by a newline produces a spurious space in the rendered post. Distasteful as it is, one must run them together: $\newcommand\…{…}$Following text. I have edited accordingly. Jun 12, 2022 at 13:05
• Thank you Loren. Old TeXing habits die hard, I had not noticed the spurious spacing. Jun 12, 2022 at 13:12
• @leomonsaingeon If the potentials are locally Lipschitz on $\Omega$, and their restrictions to $\partial \Omega$ are also locally Lipschitz, then Alexandrov theorem (applied to $\Omega$, $\partial \Omega$) would give a.e. uniqueneness with respect to $\mathcal{L}^d|_\Omega$ and $\mathcal{L}^{d-1}|_{\partial \Omega}$. I'm sure that's obvious to you, and therefore the gradients of the max Kantorovich potentials are a.e. uniquely defined by the OT. But pointwise uniqueness of $\phi, \psi$ is more difficult, and I don't know any references.
– JHM
Jun 13, 2022 at 11:29

Uniqueness of Kantorovich potentials (up to a constant shift) has been analyzed in a very general framework in this work of ours: "On the Uniqueness of Kantorovich Potentials" - https://arxiv.org/pdf/2201.08316.pdf .

Your setting is encompassed in Corollary 2. The key idea is that by optimality, the gradient of the Kantorovich potential is uniquely determined on a subset of $$\text{int}(\Omega)$$ with full Lebesgue measure. Since for your setting, the Kantorovich potential is locally Lipschitz on $$\Omega$$ it follows that it is uniquely characterized on the (connected) domain $$\Omega$$. Let me emphasize that there is no need for $$\mu$$ or $$\nu$$ to be absolutely continuous, uniqueness rather depends on the topology of the support of the underlying measures.

Our work also provides sufficient conditions for Kantorovich potentials to be unique if the measures have disconnected support. This could be helpful to extend the scope of your work.

• Lemma 7 of your paper is interesting. Do you find Qi's proof satisfactory?
– JHM
Jun 18, 2022 at 23:25
• I agree that the argument of Qi (1989) might appear a bit quick but we did confirm its validity. Jun 19, 2022 at 9:13
• Thanks @ShayanHundrieser for your helpful input. However, it still seems to me that 1) uniqueness only holds for c-concave potentials (could there be a pathological scenario where 2 optimal potentials exist, one being c-concave and the other not?), and 2) that you still need somehow that my $\mu$ measure has full support (as in F. Santambrogio's prop. 7.18). For example your corollary 2 fails if my measure $\mu$ is only supported on the boundary $\partial\Omega$, does it not? Jun 19, 2022 at 20:50
• @Leomonsaingeon: Thank you for your questions. 1) Based on Theorem 5.10(iii) in Villani (2008) you can always assume that the optimal potentials are c-concave. Moreover, for any pair of optimal potentials $f, g$, it follows that the c-transformations $(f^{cc}, f^c)$ fulfill $f^{cc}\geq f$ and $f^c \geq g$ on their whole domains. Hence, under uniqueness of c-concave potentials (up to a constant) it follows that, in fact, any pair of optimal potentials is almost surely unique (up to a constant). Jun 22, 2022 at 15:54
• 2) If $\mu$ is supported on the boundary of $\partial \Omega$, you can view $\partial \Omega$ as the underlying manifold in the first place. This way you change the topology and int(supp$(\mu))$ is non-empty. To obtain general uniqueness statements I suggest first determining the connected components of the support for your measures. Corollary 2 is a tool to verify the uniqueness on these individual components. Theorem 1 then allows you to conclude the uniqueness of Kantorovich potentials for the whole dual problem. Jun 22, 2022 at 15:54