The geometry of optimal transportation
نویسندگان
چکیده
A classical problem of transporting mass due to Monge and Kantorovich is solved. Given measures μ and ν on Rd, we find the measure-preserving map y(x) between them with minimal cost — where cost is measured against h(x − y) with h strictly 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 the Mathematical Sciences Research Institute, Berkeley, CA 94720, and RJM from the Natural Sciences and Engineering Research Council of Canada. c ©1995 by the authors. To appear in Acta Mathematica 1997.
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