# Kantorovich-Rubinstein in case of quasi-metric

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.