An optimal irrigation network with infinitely many branching points

TL;DR

This paper introduces a new convex formulation of the Gilbert-Steiner mass transportation problem, enabling the proof of optimality for complex irrigation networks with many branching points using calibration techniques.

Contribution

It presents a novel convex framework for the Gilbert-Steiner problem and demonstrates its effectiveness in proving the optimality of intricate irrigation networks.

Findings

Established a convex functional minimization approach

Developed calibration methods for optimality proofs

Proved optimality of a specific irrigation network

Abstract

The Gilbert-Steiner problem is a mass transportation problem, where the cost of the transportation depends on the network used to move the mass and it is proportional to a certain power of the "flow". In this paper, we introduce a new formulation of the problem, which turns it into the minimization of a convex functional in a class of currents with coefficients in a group. This framework allows us to define calibrations, which can be used to prove the optimality of concrete configurations. We apply this technique to prove the optimality of a certain irrigation network, having the topological property mentioned in the title.