Steiner problems in optimal transport

The problem of optimal transport, originally proposed by Monge, has a long history of investigation and application ([18] is an extensive reference). Roughly stated, the problem involves one who has an initial configuration of mass and would like to transport it to a terminal configuration of mass, doing so at least cost. For instance, one might have a set of water towers and a region of drought that one would like to relieve as quickly as possible. Abstractly, this becomes a constrained optimization problem in a space P of probability measures over the base space. Unfortunately, the mere existence of a solution is difficult to come by, due to the non-linear nature of the problem. It was over 200 years before Kantorovich [12, 11] provided serious progress by formulating and solving a weak version of the problem. We will focus on this Monge-Kantorovich problem, defined in detail below. Returning to our drought problem, suppose that the drought and even the construction of the water towers has yet to occur. The question becomes where to build the water towers to best prepare for possible droughts. If there are multiple possible droughts one wishes to protect against, but one can only afford enough water towers to combat a single drought at a time, one wishes to find a configuration of water towers which is nicely balanced amongst the possible droughts. We will investigate this by solving Steiner-type problems in the probability space P. A Steiner problem is a search for a length minimizing network, usually satisfying some boundary conditions, in a metric space. Steiner problems are traditionally solved via local compactness arguments; however, as we cannot expect local compactness from our probability space P, we will instead need to argue using the geometry of the base space. In particular, we show: