Optimal Transport from Lebesgue to Poisson

We study couplings $q^{\omega }$ of the Lebesgue measure ${\mathfrak{L}}^{\mathfrak{d}}$ 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 'optimal coupling' induces a random tiling of $\mathbb R^d$ by convex polytopes of volume 1.</p>