The Equivalence of Various Models

Abstract

This short chapter proves the equivalence of the three mathematical models generalizing the Gilbert-Steiner problem to continuous measures. The proof of this equivalence involves most structure properties proved in Chapter 7. Section 9.1 tells us in short that the Gilbert-Steiner and the optimal traffic plan problems are equivalent for finite atomic measures. Section 9.2 is dedicated to the equivalence of the traffic plan model with the Maddalena et al. irrigation pattern model [59]. Finally Section 9.3 treats the equivalence between traffic plans and Xia’s tranport paths. In all cases the equivalence is strong, as not only the minimal energies are the same but the minimizing objects, of very diverse nature, can be identified with each other. The presentation here follows [13]. Several tools come from [58] and [70].