The Structure of Optimal Traffic Plans

Abstract

This chapter gives a more geometric form to traffic plans and their energy which was first proved in [59], [11]. Section 4.1 shows that all arcs of a traffic plan can be parameterized by length without changing the energy. Section 4.3.1 shows that traffic plans with finite energy are countably rectifiable. This first regularity result is used in Section 4.3 to obtain the sonsistency of the traffic plan energy $\varepsilon ^\alpha$ with the original Gilbert energy. One has

$$ \varepsilon ^\alpha \left( P \right) = E^\alpha \left( P \right): = \int_{\mathbb{R}^N } {\left| x \right|_\chi ^\alpha d\mathcal{H}^1 \left( x \right)} $$

((4.1))

for simple paths traffic plans. Section 4.2 gives a method to actually remove the loops from any traffic plan, therefore decreasing the multiplicity |x|_χ. As apparent in (4.1) the Gilbert energy decreases with the multiplicity. Thus we get a handy and intuitive Gilbert energy formula for optimal traffic plans.