Calculus of Variations and Geometric Measure Theory

When does optimal transport branch?

Mircea Petrache (Pontificia Catolica Universidad de Chile)

created by gelli on 12 Jun 2018

20 jun 2018 -- 17:00   [open in google calendar]

Sala Seminari Dipartimento di Matematica di Pisa


Consider the problem of transporting some objects between N distinct locations. Depending on how we package together different objects and on how the transport cost (per unit of distance traveled) depends on the package that we are moving, we may cook up a minimum-cost transport strategy. Is it always the best option to let our objects travel independently of each other, or is it sometimes more cost-efficient to mergesplit packages along the way, following a branched, tree-like, global network? We consider the model in which the "packaging arithmetics" and the transport cost are quantified via a normed Abelian group G, and we extract a purely intrinsic condition on G that guarantees that the optimal transport is not branching. This seems to initiate a new geometric classification of certain normed groups. In the non-branching case we also provide global calibrations, i.e. a generalization of Monge-Kantorovich duality, by transferring the problem to a variational problem on pseudometrics. (Joint project with R. Zust)