# Is there an $\infty$ version of the Wasserstein distance between two distributions?

If I have two probability distributions $\mu$ and $\nu$ defined on $X$ and $Y$ respectively, then the $p$-th Wasserstein distance between the two of them is defined as $$W_p(\mu,\nu) = \left(\inf_{\pi\in\Pi(\mu,\nu)}\int_{X\times Y}d(x,y)^{p}\,\mathrm{d}\pi(x,y)\right)^{1/p}$$ where $\Pi(\mu,\nu)$ is the set of all distributions on $X\times Y$ whose marginals are $\mu$ and $\nu$ respectively and $p\geq 1$. My question is: is the limiting behavior of this quantity as $p\to\infty$ defined? Does it have a name?

I see no reason why $W_\infty := \inf_{\pi\in\Pi}||d||_{L^\infty(\pi)}$ wouldn't fit. This is not to claim that $\lim_{p\to\infty} W_p=W_\infty$ (which I believe is true anyway), but at least it exists, although with properties different from the $W_p$s, $p<\infty$.
• Is it clear, for instance, that your $W_\infty$ is a metric? Jul 15, 2015 at 12:32
• It is, and the limit $W_p\to W_\infty$ holds. For Polish spaces, see for instance Givens & Shortt (1984): projecteuclid.org/download/pdf_1/euclid.mmj/1029003026 Jul 15, 2015 at 12:50