I. Abraham, . Abraham, G. Bergounioux, and . Carlier, Tomographic Reconstruction from a Few Views: A Multi-Marginal Optimal Transport Approach, Applied Mathematics & Optimization, vol.24, issue.1, 2014.
DOI : 10.1007/s00245-015-9323-3

URL : https://hal.archives-ouvertes.fr/hal-01065981

M. Agueh and G. Carlier, Barycenters in the Wasserstein Space, SIAM Journal on Mathematical Analysis, vol.43, issue.2, pp.904-924, 2011.
DOI : 10.1137/100805741

URL : https://hal.archives-ouvertes.fr/hal-00637399

L. Ambrosio, N. Gigli, and G. Savaré, Gradient flows: in metric spaces and in the space of probability measures, 2006.

M. Beiglböck and C. Griessler, An optimality principle with applications in optimal transport. arXiv preprint, 2014.

M. Beiglböck, P. Henry-labordère, and F. Penkner, Model-independent bounds for option prices???a mass transport approach, Finance and Stochastics, vol.36, issue.3, pp.477-501, 2013.
DOI : 10.1007/s00780-013-0205-8

M. Beiglböck, C. Léonard, and W. Schachermayer, A general duality theorem for the Monge???Kantorovich transport problem, Studia Mathematica, vol.209, issue.2, 2009.
DOI : 10.4064/sm209-2-4

M. Beiglboeck, A. Cox, and M. Huesmann, Optimal transport and skorokhod embedding. arXiv preprint, 2013.

M. Beiglboeck and N. Juillet, On a problem of optimal transport under marginal martingale constraints. arXiv preprint, 2012.

J. Benamou, G. Carlier, M. Cuturi, L. Nenna, and G. Peyré, Iterative Bregman Projections for Regularized Transportation Problems, SIAM Journal on Scientific Computing, vol.37, issue.2, pp.1111-1138, 2015.
DOI : 10.1137/141000439

URL : https://hal.archives-ouvertes.fr/hal-01096124

J. Benamou, G. Carlier, and L. Nenna, A numerical method to solve optimal transport problems with coulomb cost. arXiv preprint arXiv:1505.01136, to appear as a chapter Splitting Methods in Communication and Imaging, Science and Engineering, Glowinski, S. Osher, and W. Yin, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01148954

E. Bodo, Applicazioni di meccanica quantistica: appunti per le lezioni. lecture notes for a course in quantum chemistry, Available, 2008.

G. Buttazzo, L. D. Pascale, and P. Gori-giorgi, Optimal-transport formulation of electronic density-functional theory, Physical Review A, vol.85, issue.6, p.62502, 2012.
DOI : 10.1103/PhysRevA.85.062502

A. Luis and . Caffarelli, A localization property of viscosity solutions to the monge-ampere equation and their strict convexity, Annals of Mathematics, pp.129-134, 1990.

A. Luis and . Caffarelli, Some regularity properties of solutions of monge ampere equation, Communications on pure and applied mathematics, vol.44, issue.8-9, pp.965-969, 1991.

G. Carlier and I. Ekeland, Matching for teams, Economic Theory, vol.11, issue.1, pp.397-418, 2010.
DOI : 10.1007/s00199-008-0415-z

URL : https://hal.archives-ouvertes.fr/hal-00661901

G. Carlier, V. Chernozhukov, and A. Galichon, Vector quantile regression. arXiv preprint, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01169653

G. Carlier and B. Nazaret, Optimal transportation for the determinant ESAIM: Control, Optimisation and Calculus of Variations, pp.678-698, 2008.

G. Carlier, A. Oberman, and E. Oudet, Numerical methods for matching for teams and wasserstein barycenters. arXiv preprint arXiv:1411, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00987292

H. Chen and G. Friesecke, Pair densities in density functional theory. arXiv preprint, 2015.

H. Chen, G. Friesecke, B. Christian, and . Mendl, Numerical Methods for a Kohn???Sham Density Functional Model Based on Optimal Transport, Journal of Chemical Theory and Computation, vol.10, issue.10, pp.4360-4368, 2014.
DOI : 10.1021/ct500586q

P. Chiappori, A. Galichon, and B. Salanié, The Roommate Problem: Is More Stable than You Think, SSRN Electronic Journal, 2014.
DOI : 10.2139/ssrn.1991460

P. Chiappori, J. Robert, L. P. Mccann, and . Nesheim, Hedonic price equilibria, stable matching, and optimal transport: equivalence, topology, and uniqueness, Economic Theory, vol.77, issue.1, pp.317-354, 2010.
DOI : 10.1007/s00199-009-0455-z

J. Aron, P. Cohen, W. Mori-sánchez, and . Yang, Challenges for density functional theory, Chemical Reviews, vol.112, issue.1, pp.289-320, 2011.

M. Colombo and S. D. Marino, Equality between Monge and Kantorovich multimarginal problems with Coulomb cost, Annali di Matematica Pura ed Applicata (1923 -), vol.75, issue.61, pp.1-14, 2013.
DOI : 10.1007/s10231-013-0376-0

M. Colombo and F. Stra, Counterexamples to multimarginal optimal transport maps with coulomb cost and radial measures

R. Cominetti and J. San-martín, Asymptotic analysis of the exponential penalty trajectory in linear programming, Mathematical Programming, pp.169-187, 1994.
DOI : 10.1007/BF01582220

C. Cotar, G. Friesecke, and C. Klüppelberg, Density Functional Theory and Optimal Transportation with Coulomb Cost, Communications on Pure and Applied Mathematics, vol.12, issue.3, pp.548-599, 2013.
DOI : 10.1002/cpa.21437

C. Cotar, G. Friesecke, and B. Pass, Infinite-body optimal transport with Coulomb cost, Calculus of Variations and Partial Differential Equations, vol.59, issue.12, pp.1-26, 2013.
DOI : 10.1007/s00526-014-0803-0

M. Cuturi, Sinkhorn distances: Lightspeed computation of optimal transport, Advances in Neural Information Processing Systems, pp.2292-2300, 2013.

L. De and P. , Optimal transport with coulomb cost. approximation and duality. arXiv preprint, 2015.

S. Di-marino, L. D. Pascale, and M. Colombo, Multimarginal optimal transport maps for 1-dimensional repulsive costs, CANADIAN JOURNAL OF MATHEMATICS- JOURNAL CANADIEN DE MATHEMATIQUES, vol.67, pp.350-368, 2015.

S. Di-marino, A. Gerolin, L. Nenna, M. Seidl, and P. Gori-giorgi, The strictly-correlated electron functional for spherically symmetric systems revisited

Y. Dolinsky and H. Soner, Martingale optimal transport and robust hedging in continuous time. Probability Theory and Related Fields, pp.391-427, 2014.

Y. Dolinsky and H. Soner, Robust hedging with proportional transaction costs, Finance and Stochastics, vol.5, issue.2, pp.327-347, 2014.
DOI : 10.1007/s00780-014-0227-x

I. Ekeland, An optimal matching problem ESAIM: Control, Optimisation and Calculus of Variations, pp.57-71, 2005.

C. Lawrence, M. Evans, and . Zworski, Lectures on semiclassical analysis, 2007.

G. Friesecke, The multiconfiguration equations for atoms and molecules: charge quantization and existence of solutions. Archive for Rational Mechanics and Analysis, pp.35-71, 2003.

G. Friesecke, B. Christian, B. Mendl, C. Pass, C. Cotar et al., N-density representability and the optimal transport limit of the Hohenberg-Kohn functional, The Journal of Chemical Physics, vol.139, issue.16, p.139164109, 2013.
DOI : 10.1063/1.4821351

A. Galichon, P. Henry-labordère, and N. Touzi, A stochastic control approach to no-arbitrage bounds given marginals, with an application to lookback options, The Annals of Applied Probability, vol.24, issue.1, pp.312-336, 2014.
DOI : 10.1214/13-AAP925

A. Galichon and B. Salanié, Matching with Trade-Offs: Revealed Preferences Over Competing Characteristics, SSRN Electronic Journal, 2010.
DOI : 10.2139/ssrn.1487307

W. Gangbo and A. Swiech, Optimal maps for the multidimensional Monge-Kantorovich problem, Communications on Pure and Applied Mathematics, vol.51, issue.1, pp.23-45, 1998.
DOI : 10.1002/(SICI)1097-0312(199801)51:1<23::AID-CPA2>3.0.CO;2-H

N. Ghoussoub and B. Maurey, Remarks on multi-marginal symmetric mongekantorovich problems. arXiv preprint, 2012.

N. Ghoussoub and A. Moameni, Symmetric monge?kantorovich problems and polar decompositions of vector fields. Geometric and Functional Analysis, pp.1129-1166, 2014.

N. Ghoussoub and B. Pass, Decoupling of DeGiorgi-Type Systems via Multi-Marginal Optimal Transport, Communications in Partial Differential Equations, vol.10, issue.6, pp.1032-1047, 2014.
DOI : 10.1002/cpa.3160380515

P. Gori-giorgi, M. Seidl, and G. Vignale, Density-functional theory for strongly interacting electrons. Physical review letters, p.166402, 2009.

L. Kantorovich, On a Problem of Monge, Journal of Mathematical Sciences, vol.133, issue.4, pp.1383-1383, 2006.
DOI : 10.1007/s10958-006-0050-9

L. Kantorovitch, On the Translocation of Masses, Management Science, vol.5, issue.1, pp.1-4, 1958.
DOI : 10.1287/mnsc.5.1.1

G. Hans and . Kellerer, Duality theorems for marginal problems, pp.399-432, 1984.

Y. Kim and B. Pass, Multi-marginal optimal transport on riemannian manifolds . arXiv preprint, 2013.

Y. Kim and B. Pass, A General Condition for Monge Solutions in the Multi-Marginal Optimal Transport Problem, SIAM Journal on Mathematical Analysis, vol.46, issue.2, pp.1538-1550, 2014.
DOI : 10.1137/130930443

Y. Kim and B. Pass, Wasserstein barycenters over riemannian manfolds. arXiv preprint, 2014.

J. Kitagawa and B. Pass, The multi-marginal optimal partial transport problem. arXiv preprint, 2014.

M. Knott and C. S. Smith, On a generalization of cyclic monotonicity and distances among random vectors. Linear algebra and its applications, pp.363-371, 1994.

V. Alexander, . Kolesnikov, A. Danila, and . Zaev, Optimal transportation of processes with infinite kantorovich distance. independence and symmetry. arXiv preprint, 2013.

C. David, . Langreth, P. John, and . Perdew, The exchange-correlation energy of a metallic surface, Solid State Communications, vol.17, issue.11, pp.1425-1429, 1975.

H. Elliott and . Lieb, A lower bound for coulomb energies, Physics Letters A, vol.70, issue.5, pp.444-446, 1979.

H. Elliott and . Lieb, Density functionals for coulomb systems, Inequalities, pp.269-303, 2002.

H. Elliott, S. Lieb, and . Oxford, Improved lower bound on the indirect coulomb energy, International Journal of Quantum Chemistry, vol.19, issue.3, pp.427-439, 1981.

H. Elliott, B. Lieb, and . Simon, The hartree-fock theory for coulomb systems, Communications in Mathematical Physics, vol.53, issue.3, pp.185-194, 1977.

P. Lions, Solutions of Hartree-Fock equations for Coulomb systems, Communications in Mathematical Physics, vol.22, issue.1, pp.33-97, 1987.
DOI : 10.1007/BF01205672

O. Artur, . Lopes, K. Jairo, and . Mengue, Duality theorems in ergodic transport, Journal of Statistical Physics, vol.149, issue.5, pp.921-942, 2012.

O. Artur, . Lopes, R. Elismar, P. Oliveira, and . Thieullen, The dual potential, the involution kernel and transport in ergodic optimization, 2011.

F. Malet and P. Gori-giorgi, Strong Correlation in Kohn-Sham Density Functional Theory, Physical Review Letters, vol.109, issue.24, p.246402, 2012.
DOI : 10.1103/PhysRevLett.109.246402

B. Christian, L. Mendl, and . Lin, Kantorovich dual solution for strictly correlated electrons in atoms and molecules, Physical Review B, vol.87, issue.12, p.125106, 2013.

G. Monge, Mémoire sur la théorie des déblais et des remblais. De l'Imprimerie Royale, p.1781

I. Olkin and S. T. Rachev, Maximum Submatrix Traces for Positive Definite Matrices, SIAM Journal on Matrix Analysis and Applications, vol.14, issue.2, pp.390-397, 1993.
DOI : 10.1137/0614027

B. Pass, Structural results on optimal transportation plans, 2011.

B. Pass, Uniqueness and Monge Solutions in the Multimarginal Optimal Transportation Problem, SIAM Journal on Mathematical Analysis, vol.43, issue.6, pp.2758-2775, 2011.
DOI : 10.1137/100804917

B. Pass, On the local structure of optimal measures in the multi-marginal optimal transportation problem, Calculus of Variations and Partial Differential Equations, vol.40, issue.1, pp.3-4529, 2012.
DOI : 10.1007/s00526-011-0421-z

B. Pass, Remarks on the semi-classical Hohenberg???Kohn functional, Nonlinearity, vol.26, issue.9, p.2731, 2013.
DOI : 10.1088/0951-7715/26/9/2731

B. Pass, Multi-marginal optimal transport and multi-agent matching problems: Uniqueness and structure of solutions, Discrete and Continuous Dynamical Systems, vol.34, issue.4, pp.1623-1639, 2014.
DOI : 10.3934/dcds.2014.34.1623

B. Pass, Multi-marginal optimal transport: theory and applications. arXiv preprint, 2014.

S. Paziani, S. Moroni, P. Gori-giorgi, and G. B. Bachelet, Local-spin-density functional for multideterminant density functional theory, Physical Review B, vol.73, issue.15, p.155111, 2006.
DOI : 10.1103/PhysRevB.73.155111

A. Pratelli, On the equality between Monge's infimum and Kantorovich's minimum in optimal mass transportation, Annales de l'Institut Henri Poincare (B) Probability and Statistics, pp.1-13, 2007.
DOI : 10.1016/j.anihpb.2005.12.001

A. Pratelli, On the sufficiency of c-cyclical monotonicity for optimality of transport plans, Mathematische Zeitschrift, vol.27, issue.2, 2007.
DOI : 10.1007/s00209-007-0191-7

D. Robert, Autour de l'approximation semi-classique, Birkhäuser Basel, vol.68, 1987.

L. Ruschendorf, Convergence of the iterative proportional fitting procedure. The Annals of Statistics, pp.1160-1174, 1995.

L. Rüschendorf and W. Thomsen, Closedness of sum spaces andthe generalized schrödinger problem. Theory of Probability & Its Applications, pp.483-494, 1998.

L. Rüschendorf and L. Uckelmann, On Optimal Multivariate Couplings, 1997.
DOI : 10.1007/978-94-011-5532-8_31

W. Schachermayer and J. Teichmann, Characterization of optimal transport plans for the Monge-Kantorovich problem, Proceedings of the American Mathematical Society, vol.137, issue.02, pp.519-529, 2009.
DOI : 10.1090/S0002-9939-08-09419-7

E. Schrödinger, ¨ Uber die umkehrung der naturgesetze. Verlag Akademie der wissenschaften in kommission bei, 1931.

M. Seidl, Strong-interaction limit of density-functional theory, Physical Review A, vol.60, issue.6, p.4387, 1999.
DOI : 10.1103/PhysRevA.60.4387

M. Seidl, P. Gori-giorgi, and A. Savin, Strictly correlated electrons in density-functional theory: A general formulation with applications to spherical densities, Physical Review A, vol.75, issue.4, p.42511, 2007.
DOI : 10.1103/PhysRevA.75.042511

C. Smith and M. Knott, On Hoeffding-Fr??chet bounds and cyclic monotone relations, Journal of Multivariate Analysis, vol.40, issue.2, pp.328-334, 1992.
DOI : 10.1016/0047-259X(92)90029-F

J. Toulouse, F. Colonna, and A. Savin, Short-range exchange and correlation energy density functionals: Beyond the local-density approximation, The Journal of Chemical Physics, vol.122, issue.1, p.14110, 2005.
DOI : 10.1063/1.1824896

URL : https://hal.archives-ouvertes.fr/hal-00981845

C. Villani, Optimal transport: old and new, 2008.
DOI : 10.1007/978-3-540-71050-9

G. Xia, S. Ferradans, G. Peyré, and J. Aujol, Synthesizing and Mixing Stationary Gaussian Texture Models, SIAM Journal on Imaging Sciences, vol.7, issue.1, pp.476-508, 2014.
DOI : 10.1137/130918010

URL : https://hal.archives-ouvertes.fr/hal-00816342

W. Yang, Generalized adiabatic connection in density functional theory, The Journal of Chemical Physics, vol.109, issue.23, pp.10107-10110, 1998.
DOI : 10.1063/1.477701

D. Zaev, On the monge-kantorovich problem with additional linear constraints. arXiv preprint, 2014.

M. Grigorii and . Zhislin, Discussion of the spectrum of schrödinger operators for systems of many particles, Trudy Moskovskogo matematiceskogo obscestva, vol.9, pp.81-120, 1960.