Timeline for Is there an analogue of transportation-cost inequality under a weighted Log-Sobolev Inequality?
Current License: CC BY-SA 4.0
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| May 20, 2021 at 16:08 | history | edited | leo monsaingeon |
added the optimal-transportation tag
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| May 20, 2021 at 6:21 | answer | added | leo monsaingeon | timeline score: 1 | |
| May 19, 2021 at 20:13 | comment | added | leo monsaingeon | I see... then I suspect this is really tricky, getting sharp constants for such highly nonlinear functional inequalities can be delicate. One last comment, though: in the Euclidean setting the $\beta/2$ factor is related to displacement convexity (in the sense of McCann) of the relative entropy $\rho\mapsto H(\rho|\pi)$, which in turn is related to log-concavity of the reference measure $\pi$. There are results about displacement convexity over manifolds, in which case the Ricci curvature plays a significant role. Perhaps it's worth looking into it? (I recommend Villani's "big book") | |
| May 19, 2021 at 20:09 | comment | added | user_qj | Thanks yes I understand the comment - but don't really want to reduce to the Euclidean setting in the bound :( | |
| May 19, 2021 at 19:55 | comment | added | leo monsaingeon | Well, certainly if your Riemannian metric tensor $G(x)$ is uniformly bounded from below and from above you will get a similar inequality by paying a factor $C\approx (\lambda/\Lambda)^2$ or $C\approx (\Lambda/\lambda)^2$ in front of $\frac\beta 2$, where $\lambda\succ G(x)\succ \Lambda$ are the lower and upper bounds. In other words you can use the fact that the inequality holds true in the Euclidean setting. But perhaps you are really interested in the sharp $\frac{\beta}{2}$ factor? (in which case I don't know the answer, but I suspect this is actually not trivial at all) | |
| May 19, 2021 at 18:22 | history | edited | user_qj |
edited tags
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| May 19, 2021 at 18:14 | history | edited | user_qj |
edited tags
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| May 19, 2021 at 18:08 | review | First posts | |||
| May 19, 2021 at 18:33 | |||||
| May 19, 2021 at 18:06 | history | asked | user_qj | CC BY-SA 4.0 |