Optimal transportation networks : models and theory / Marc Bernot, Vicent Caselles, Jean-Michel Morel

The authors provide a rigorous treatment and mathematical sound foundation of a bunch of models and aspects concerning the design and analysis of transportation networks. They present a general framework where wellknown models (Monge-Kantorovich, Gilbert-Steiner), more recent approaches (Patterns/fiber trees, Traffic Plans for irrigation problems as well as for problems with additionally “who goes where" constraints) and specific models f.e. from geophysics (joint landscape-river network evolution) are incorporated in a common mathematical structure, analysed and compared. Initial motivation stems from the observation that transportation costs usually are subadditive and increasing with respect to the flow value v; the simplest model therefore assumes exponential flowcost with an exponent chosen from the interval (0,1). Often claimed structural characteristics of (cost/energy-)optimal transport schemes: single path property, trunk trees decomposition, finite straight graph approximability, global bound on the number of branches etc. are discussed in depth and clearified. Equivalence of the main models is proven for specific scenarios; the existence of a traffic plan - to transport a positive Borel measure on the N-dimensional real space into another one with finite energy - is shown to be linked with the Hausdorff and Minkowski dimensions of the support of the measures. A list of interesting open problems and conjectures concludes this expose. (Zentralblatt)