Timeline for Uniqueness of Kantorovich potentials?
Current License: CC BY-SA 4.0
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| Dec 20, 2022 at 8:37 | comment | added | Akira | @JHM Some months ago, I encountered the paper Determination of functions by metric slopes which may be of your interest... | |
| Jun 22, 2022 at 15:54 | comment | added | Shayan Hundrieser | 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 | comment | added | Shayan Hundrieser | @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 19, 2022 at 20:50 | comment | added | leo monsaingeon | 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 9:13 | comment | added | Shayan Hundrieser | I agree that the argument of Qi (1989) might appear a bit quick but we did confirm its validity. | |
| Jun 18, 2022 at 23:25 | comment | added | JHM | Lemma 7 of your paper is interesting. Do you find Qi's proof satisfactory? | |
| Jun 18, 2022 at 22:23 | history | edited | Shayan Hundrieser | CC BY-SA 4.0 |
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| S Jun 18, 2022 at 22:22 | review | First answers | |||
| Jun 18, 2022 at 22:36 | |||||
| S Jun 18, 2022 at 22:22 | history | answered | Shayan Hundrieser | CC BY-SA 4.0 |