Optimal transportation, topology and uniqueness

Details Optimal transportation, topology and uniqueness Optimal transportation, topology and uniqueness

2010-08-26

arxiv.org/abs/1008.4419v1

Mathematics

Probability

36 pages, 6 figures

Scientific paper

The Monge-Kantorovich transportation problem involves optimizing with respect to a given a cost function. Uniqueness is a fundamental open question about which little is known when the cost function is smooth and the landscapes containing the goods to be transported possess (non-trivial) topology. This question turns out to be closely linked to a delicate problem (# 111) of Birkhoff [14]: give a necessary and sufficient condition on the support of a joint probability to guarantee extremality among all measures which share its marginals. Fifty years of progress on Birkhoff's question culminate in Hestir and Williams' necessary condition which is nearly sufficient for extremality; we relax their subtle measurability hypotheses separating necessity from sufficiency slightly, yet demonstrate by example that to be sufficient certainly requires some measurability. Their condition amounts to the vanishing of the measure \gamma outside a countable alternating sequence of graphs and antigraphs in which no two graphs (or two antigraphs) have domains that overlap, and where the domain of each graph / antigraph in the sequence contains the range of the succeeding antigraph (respectively, graph). Such sequences are called numbered limb systems. We then explain how this characterization can be used to resolve the uniqueness of Kantorovich solutions for optimal transportation on a manifold with the topology of the sphere.

Affiliated with

Also associated with

No associations

Rating

Optimal transportation, topology and uniqueness does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Optimal transportation, topology and uniqueness, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Optimal transportation, topology and uniqueness will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-703603