Abstract for the talk at 17.04.2012 (15:15 h)

Oberseminar ANALYSIS
Karl-Theodor Sturm (Universität Bonn)
Optimal Transport from Lebesgue to Poisson
We study couplings q^ω of the Lebesgue measure 𝔏^d and the Poisson point process μ^ω on 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 ∈ (0,∞) provided d ≥ 3. If d ≤ 2 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 ℝ^d by convex polytopes of volume 1.