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Consider the classical Kantorovich-Rubinstein Duality Theorem, \begin{align*} \inf_{\{\pi \mid \pi|_{X\times pt}=\mu, \pi|_{pt\times X}=\nu \}} \int_{X^2} d(x,y)\pi(x,y) = \sup_{l\in Lip_1(X)} \int_X l(x) (\mu-\nu)(x). \end{align*} Int the finite case the proof can be reduced to linear programming duality and the fact that for a metric $d$ the $d$-concave dual of a 1-Lipschitz function $f$ satisfies $f^{dd}=-f^d$. The second condition does not hold in the case of a quasi-metric (symmetry condition from the definition of a metric is omitted). My questions are:

  1. Does KR hold for quasi-metric?

  2. If not, what KR-type result holds for quasi-metric?

I would be grateful for any references. I saw some people mentioning some KR-type results obtained by Hutchinson but was unable to get the full references.

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See Section 4 of:

https://ac.els-cdn.com/0166218X93E0139P/1-s2.0-0166218X93E0139P-main.pdf?_tid=570ec60e-b559-11e7-9ef9-00000aab0f6c&acdnat=1508478277_c4761bc77418e888965b6d6c57a90283

The paper is about countable linear programming, but that gives you exactly what you need in the countable discrete case (namely that you need symmetry).

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