Timeline for Monge-Kantorovich duality with a $\{0,1\}$ cost function
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 4, 2019 at 18:57 | vote | accept | Tom Solberg | ||
| Mar 28, 2019 at 14:35 | answer | added | Martin Kell | timeline score: 2 | |
| Mar 27, 2019 at 20:24 | history | edited | Tom Solberg | CC BY-SA 4.0 |
added an additional assumption
|
| Mar 27, 2019 at 20:08 | comment | added | Tom Solberg | @Dirk yes the theorem I'm looking at (Theorem 1.3 of Villani's "Topics in Optimal Transporation") says that you can assume WLOG that the potentials are continuous, but we're taking a supremum as opposed to a maximum. I'm updating the question with some additional findings now. | |
| Mar 27, 2019 at 19:37 | comment | added | Dirk | I think Steve meant that the inequality should hold everywhere (aren't the potentials continuous functions anyway?). | |
| Mar 27, 2019 at 19:35 | comment | added | Dirk | The fundamental theorem of optimal transport says that any measure on the product space that has its support in the c-superdifferential of a c-concave function is optimal for it's marginals. So you can try to build a counterexample by finding a phi which has other values than 1, 0, -1 and which is also c-concave. | |
| Mar 27, 2019 at 19:30 | history | edited | Tom Solberg | CC BY-SA 4.0 |
clarified "almost all"
|
| Mar 27, 2019 at 19:29 | comment | added | Tom Solberg | Thanks @Steve! I made a correction regarding your second comment. | |
| Mar 27, 2019 at 18:53 | comment | added | Steve | Two quick comments: First, I posted a wrong answer earlier where I simply messed up a detail, sorry! Second should "$\varphi(x) + \psi(y) \leq c(x, y)$ for almost all ..." be instead pointwise (since no measure on the product space is given)? | |
| Mar 26, 2019 at 18:04 | history | asked | Tom Solberg | CC BY-SA 4.0 |