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 qω of the Lebesgue measure 𝔏d and the Poisson point process μω on ℝd. We ask for a minimizer of the mean Lp-transportation cost. The minimal mean Lp-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 Lp-transportation cost. In the case p = 2, this ’optimal coupling’ induces a random tiling of ℝd by convex polytopes of volume 1.