A classical problem of transporting mass due to Monge and Kantorovich is solved. Given measures µ and ν on R d, we find the measure-preserving map y(x) between them with minimal cost — where cost is measured against h(x − y) withhstrictly convex, or a strictly concave function of |x − y|. This map is unique: it is characterized by the formula y(x) =x−(∇h) −1 (∇ψ(x)) and geometrical restrictions on ψ. Connections with mathematical economics, numerical computations, and the Monge-Ampère equation are sketched. ∗ Both authors gratefully acknowledge the support provided by postdoctoral fellowships: WG at
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Gangbo et al. (1996) studied this question.
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