Optimal transportation networks as flat chains

Abstract

We provide a model of optimization of transportation networks (e.g.\ urban traffic lines, subway or railway networks) in a geografical area (e.g. a city) with given density of population and that of services and/or workplaces, the latter being the destinations of everyday movements of the former. The model is formulated in terms of Federer-Fleming theory of currents, and allows to get both the position and the necessary capacity of the optimal network. Existence and some qualitative properties of solutions to the respective optimization problem are studied. Also, in an important particular case it is shown that the model proposed is equivalent to another known model of optimization of optimal transportation network, the latter not using the language of currents.