Optimal transportation networks and stations

Here c is a given lower semicontinuous cost function, and πi are the marginals of π . In the typical case c(x, y) = ‖x − y‖, p > 1, and the pth root of the minimum in (1.1) is then called the p-Wasserstein distance Wp(μ, μ) between μ and μ. Brancolini and Buttazzo [6], Buttazzo and Stepanov [8], and Buttazzo et al. [7], among other works, consider a variant of the problem with two different media. The object is then to find a onedimensional network or track Σ that attains (1.1) when the cost c of travelling between x and y is defined in such a way that it is cheaper to travel on Σ than outside it. Another variant of the problem is considered by Bernot et al. [5], and Xia [17, 18, 19], among others. Although employing a single medium, this version also attempts to include the cost of building the transportation network by considering a target functional that attempts to group together the paths taken by mass travelling from μ to μ, forming a rectifiable network. In the language of Federer–Fleming currents, this can be formulated as finding a current T of optimal weighted area M(T ) (α ∈ [0, 1)) such that ∂T = μ. Paolini and Stepanov [14] also study a variant of the two-media problem in the language of currents: that of minimising the functional