Optimal Transport from Lebesgue to Poisson

We study couplings $q^\omega$ of the Lebesgue measure $\frak L^d$ and the Poisson point process $\mu^\omega$ on $\mathbb R^d$. We ask for a minimizer of the mean $L^p$-transportation cost.

The minimal mean $L^p$-transportation cost turns out to be finite for all $p\in (0,\infty)$ provided $d\ge3$. If $d\le2$ then it is finite if and only if $p<d/2$.

Moreover, in any of these cases we prove that there exist a unique translation invariant coupling which minimizes the mean $L^p$-transportation cost. In the case $p=2$, this &#039;optimal coupling&#039; induces a random tiling of $\mathbb R^d$ by convex polytopes of volume 1.</p>