Dirac to Lebesgue Segment: A Case Study

Abstract

In this chapter, we investigate the structure of an optimal traffic plan irrigating the Lebesgue measure on the segment from a single source. In the case of Monge-Kantorovich transport, as illustrated by Figure 13.1, an optimal traffic plan is totally spread in the sense that fibers connect every point of the segment with the source. If α=0, which corresponds to the problem of Steiner, an optimal traffic plan is such that all the mass is first conveyed to some point of the segment and then sent onto the whole segment. What is the shape of an optimal traffic plan for an intermediate α ∈ (0,1)? This is the question we explore in this chapter, both from an analytical and experimental point of view. In the first section, we first prove in Lemma 13.6 that an optimal traffic plan cannot be like the one for α=0, i.e. with no bifurcation away from the segment (configuration that we call T—structure). Indeed, we prove by a perturbation argument that there exists a traffic plan with a Y—structure that has a lower cost than the T—structure one. This implies that there have to be an infinite number of bifurcations (see Corollary 13.11). Indeed, if there were a finite number of bifurcations, we could extract an optimal traffic plan with T—structure (see Figure 13.5). With the same perturbation technique, we finally prove in Proposition 13.12 that whenever the traffic plan conveys some positive mass on the segment, the path that accomplishes this transportation has to be tangent to the segment.