When does optimal transport branch?

Wednesday, June 20, 2018
Ora Inizio: 
17:00
Ora Fine: 
18:00
Mircea Petrache
Pontificia Universidad Católica de Chile

Abstract. 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 merge/split
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)