On the local structure of optimal measures in the multi-marginal optimal transportation problem

We consider an optimal transportation problem with more than two marginals. We use a family of semi-Riemannian metrics derived from the mixed, second order partial derivatives of the cost function to provide upper bounds for the dimension of the support of the solution.

[1]  W. Gangbo,et al.  The geometry of optimal transportation , 1996 .

[2]  M. Knott,et al.  On Hoeffding-Fre´chet bounds and cyclic monotone relations , 1992 .

[3]  L. Kantorovich.  On the Translocation of Masses , 2006 .

[4]  G. Carlier,et al.  Matching for teams , 2010 .

[5]  Brendan Pass,et al.  Uniqueness and Monge Solutions in the Multimarginal Optimal Transportation Problem , 2010, SIAM J. Math. Anal..

[6]  W. Gangbo,et al.  Optimal maps for the multidimensional Monge-Kantorovich problem , 1998 .

[7]  Guillaume Carlier,et al.  Barycenters in the Wasserstein Space , 2011, SIAM J. Math. Anal..

[8]  V. Levin.  Abstract Cyclical Monotonicity and Monge Solutions for the General Monge–Kantorovich Problem , 1999 .

[9]  M. Knott,et al.  On a generalization of cyclic monotonicity and distances among random vectors , 1994 .

[10]  Henri Heinich,et al.  Problème de Monge pour probabilités , 2002 .

[11]  P. Chiappori,et al.  Hedonic price equilibria, stable matching, and optimal transport: equivalence, topology, and uniqueness , 2007 .

[12]  L. Rüschendorf,et al.  On Optimal Multivariate Couplings , 1997 .

[13]  Guillaume Carlier,et al.  OPTIMAL TRANSPORTATION FOR THE DETERMINANT , 2006, math/0612142.

[14]  Guillaume Carlier,et al.  On a Class of Multidimensional Optimal Transportation Problems , 2003 .

[15]  R. McCann,et al.  Rectifiability of Optimal Transportation Plans , 2010, Canadian Journal of Mathematics.

[16]  H. Kellerer.  Duality theorems for marginal problems , 1984 .

[17]  R. McCann,et al.  Continuity, curvature, and the general covariance of optimal transportation , 2007, 0712.3077.

[18]  R. McCann.  Polar factorization of maps on Riemannian manifolds , 2001 .

[19]  Svetlozar T. Rachev,et al.  Maximum submatrix traces for positive definite matrices , 1993 .

[20]  L. Rüschendorf,et al.  On the n-Coupling Problem , 2002 .

[21]  G. Minty.  Monotone (nonlinear) operators in Hilbert space , 1962 .