# The oriented mailing problem and its convex relaxation

## TL;DR

This paper introduces an oriented model for the mailing problem in branched transportation, incorporating particle orientation into the cost functional, and proposes convex relaxations with calibration for related network optimization problems.

## Contribution

It presents a novel oriented mailing model and convex relaxation framework with calibration, extending existing transportation network theories.

## Key findings

- Effective solution to Problem 15.9 in 'Optimal transportation networks'
- Convex relaxation using rectifiable currents with group coefficients
- Calibration concept for the new model and related Steiner tree variant

## Abstract

In this note we introduce a new model for the mailing problem in branched transportation in order to allow the cost functional to take into account the orientation of the moving particles. This gives an effective answer to [Problem 15.9] of the book "Optimal transportation networks" by Bernot, Caselles, and Morel. Moreover we define a convex relaxation in terms of rectifiable currents with group coefficients. With such approach we provide the problem with a notion of calibration. Using similar techniques we define a convex relaxation and a corresponding notion of calibration for a variant of the Steiner tree problem in which a connectedness constraint is assigned only among a certain partition of a given set of finitely many points.