The geometry of optimal transportation
1996/01/01 by Wilfrid Gangbo, Robert J. McCann · 867 citations
Mathematics · #Geometric Analysis and Curvature Flows #Geometry #Mathematics #Mathematics and Applications #Point processes and geometric inequalities
paper · pdf · doi:10.1007/bf02392620
published in Acta Mathematica 177(2), 113-161 (Mittag-Leffler Institute)
openalex publication_date 1996/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Abstract
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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