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The geometry of optimal transportation. (English) Zbl 0887.49017
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 $$C_1(s)= \int c(x,s(x))d\mu(x)$$. 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 $$s$$. 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.

##### MSC:
 49J99 Existence theories in calculus of variations and optimal control 91B60 Trade models 90B06 Transportation, logistics and supply chain management
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