Comparison of distances between measures

Abstract

The problem of optimal transportation between a set of sources and a set of wells has become recently the object of new mathematical models generalizing the Monge–Kantorovich problem. These models are more realistic as they predict the observed branching structure of communication networks. They also define new distances between measures. The question arises of how these distances compare to the classical Wasserstein distance obtained from the Monge–Kantorovich problem. In this work we show sharp inequalities between the $d_{\alpha }$ distance induced by branching transport paths and the classical Wasserstein distance over probability measures in a compact domain of ${\mathbb{R}}^m$.