Traffic Plans and Distances between Measures

Abstract

In this chapter, we consider the irrigation and who goes where problems for the cost functional E^α introduced at the end of Chapter 3. We prove in Section 6.1 that for $\alpha > 1 - \tfrac{1} {N}$ where N is the dimension of the ambient space, the optimal cost to transport µ⁺ to μ⁻ is finite. More precisely, if μ⁺ and μ⁻ are two nonnegative measures on a domain X with the same total mass M and $\alpha > 1 - 1/N$, set

$$ E^\alpha \left( {\mu ^ + ,\mu ^ - } \right): = \mathop {\min }\limits_{\chi \in TP\left( {\mu ^ + ,\mu ^ - } \right)} E^\alpha \left( \chi \right). $$

((6.1))

Then (μ⁺,μ⁻ can be bounded by

$$ E^\alpha \left( {\mu ^ + ,\mu ^ - } \right) \leqslant C_{\alpha ,N} M^\alpha diam\left( X \right). $$

The proof of this property, first proven in [94], follows from the explicit construction of a dyadic tree connecting any probability measure on X to a Dirac mass. If α is under this threshold it may happen that the iofimum is in fact ∞.