The Steiner tree problem revisited through rectifiable G-currents

TL;DR

This paper reformulates the classical Steiner tree problem as a mass-minimization problem for 1-dimensional currents with coefficients in a normed group, enabling new analytical approaches and calibration techniques.

Contribution

It introduces a novel formulation of the Steiner tree problem using rectifiable G-currents and establishes a calibration principle for this framework.

Findings

Calibration examples demonstrate the effectiveness of the approach

New formulation provides analytical tools for Steiner tree problem

Potential for improved solution techniques in geometric optimization

Abstract

The Steiner tree problem can be stated in terms of finding a connected set of minimal length containing a given set of finitely many points. We show how to formulate it as a mass-minimization problem for $1$-dimensional currents with coefficients in a suitable normed group. The representation used for these currents allows to state a calibration principle for this problem. We also exhibit calibrations in some examples.