Abstract
We develop and study a theory of optimal transport for vector measures. We resolve in the negative a conjecture of Klartag, that given a vector measure on Euclidean space with total mass zero, the mass of any transport set is again zero. We provide a counterexample to the conjecture. We generalise the Kantorovich–Rubinstein duality to the vector measures setting. Employing the generalisation, we answer the conjecture in the affirmative provided there exists an optimal transport with absolutely continuous marginals of its total variation.
Highlights
In this note we develop theory of optimal transport of vector measures
Let us first briefly describe the topic of classical optimal transport
The localisation technique stems from convex geometry, but its generalisations have been employed to prove many novel results concerning functional inequalities, e.g. isoperimetric inequality in the metric measure spaces satisfying the synthetic curvature-dimension condition
Summary
In 1781 Gaspard Monge (see [19]) asked the following question: given two probability distributions μ, ν on a metric space (X , d), how to transfer one distribution onto the other in an optimal way. The localisation technique stems from convex geometry, but its generalisations have been employed to prove many novel results concerning functional inequalities, e.g. isoperimetric inequality in the metric measure spaces satisfying the synthetic curvature-dimension condition (see [7,8]). The latter notion was introduced in the foundational papers by Sturm [21,22] and by Lott and Villani [18] and allowed for development of a far-reaching, vast theory of metric measure spaces. We refer the reader to [12] and references therein for a broader description of the localisation technique and its history
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
