The geometry of optimal transportation
This paper treats a classical (1781) problem of Monge whose essence is: Let \((X,\mu)\) and \((Y,\nu)\) be (probability) measure spaces, where \(\mu\) represents the distribution of the production of some commodity at various ``localities'' \(x\in X\) and \(\nu\) of the consumption of that commodity at localities \(y\in Y\). There is a cost function \(c(x,y)\) on \(X\times Y\) which represents the cost of transporting the commodity from \(x\) to \(y\). A measure-preserving transformation \(s:X\to Y\) represents a ``distribution scheme'' for the commodity, resulting in a total transport cost of \(C_1(s)= \int c(x,s(x))d\mu(x)\). \textit{Monge's problem} is to find, where possible, the (or an) \(s\) which gives the minimum \(M_1\) of \(C_1(s)\) over all such choices of \(s\). \textit{Kantorovich's problem}, simpler than Monge's one, is to find, where possible, a measure \(\gamma\) on \(X\times Y\) which has \(\mu\) and \(\nu\) as marginals, and which gives the minimum \(M_2\) of the transport cost \(C_2(\gamma)= \int c(x,y)d\gamma(x, y)\) over all such choices of \(\gamma\). It is easy to see that \(M_2\leq M_1\), but there are cases where \(M_1= M_2\) and where \(s\) can be accessed through \(\gamma\). Sometimes \(s\) is even uniquely determined a.e. For their main (positive) results, the authors restrict themselves to \(X, Y\subseteq\mathbb{R}^d\), \(d= 1,2,3,\dots\), and to two classes of cost functions: (i) \(c(x,y)= h(x-y)\), where \(h\) is strictly convex (relatively easy); (ii) \(c(x,y)= l(|x-y|)\), where \(|\cdot|\) denotes Euclidean distance on \(\mathbb{R}^d\), and where \(l\geq 0\) is strictly concave (harder but more realistic in the economic context). They also give numerous examples where the (unique) map \(s\) can be explicitly constructed, and their treatment is largely self-contained.
scientific article; zbMATH DE number 1083669
scientific article