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Multi-marginal optimal transport and multi-agent matching problems: uniqueness and structure of solutions. (English) Zbl 1278.49054

Summary: We prove uniqueness and Monge solution results for multi-marginal optimal transportation problems with a certain class of surplus functions; this class arises naturally in multi-agent matching problems in economics. This result generalizes a seminal result of W. Gangbo and A. Świȩch [Commun. Pure Appl. Math. 51, No. 1, 23–45 (1998; Zbl 0889.49030)]. Of particular interest, we show that this also yields a partial generalization of the Gangbo-Świȩch result to manifolds; alternatively, we can think of this as a partial extension of R. J. McCann’s theorem for quadratic costs on manifolds [Geom. Funct. Anal. 11, No. 3, 589–608 (2001; Zbl 1011.58009)] to the multi-marginal setting. { } We also show that the class of surplus functions considered here neither contains, nor is contained in, the class of surpluses studied in [B. Pass, SIAM J. Math. Anal. 43, No. 6, 2758–2775 (2011; Zbl 1248.49061)], another generalization of Gangbo and Świȩch’s result.

MSC:

49Q20 Variational problems in a geometric measure-theoretic setting
49K20 Optimality conditions for problems involving partial differential equations
49Q15 Geometric measure and integration theory, integral and normal currents in optimization
91B68 Matching models
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