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This note exposes the differential topology and geometry underlying some of the basic phenomena of optimal transportation.
On the translocation of masses
L. Kantorovich · 1942
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An inequality for rearrangements
G.G. Lorentz · 1953
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Assignment problems and the location of economic activities
T.C. Koopmans and M. Beckmann · 1957
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Monotone (nonlinear) operators in Hilbert space
G.J. Minty · 1962
Earlier work this paper cites.
Characterization of the subdifferentials of convex functions
R.T. Rockafellar · 1966
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An exploration in the theory of optimum income taxation
J.A. Mirrlees · 1971
Earlier work this paper cites.
Job market signaling
M. Spence · 1973
Earlier work this paper cites.
Probabilities and metrics - Convergence of laws on metric spaces, with a view to statistical testing
R.M. Dudley · 1976
Earlier work this paper cites.
An extended Lagrangian model of semi-geostrophic frontogenesis
M.J.P Cullen and R.J. Purser · 1984
Earlier work this paper cites.
Décomposition polaire et réarrangement monotone des champs de vecteurs
Y. Brenier · 1987
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A necessary and sufficient condition for rationalizability in a quasi-linear context
J.-C. Rochet · 1987
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On the optimal transportation of distributions
C. Smith and M. Knott · 1987
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A characterization of random variables with minimum L 2 L^{2} -distance
L. Rüschendorf and S.T. Rachev · 1990
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Polar factorization and monotone rearrangement of vector-valued functions
Y. Brenier · 1991
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Some regularity properties of solutions of Monge-Ampère equation
L.A. Caffarelli · 1991
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Classical solvability in dimension two of the second boundary-value problem associated with the Monge-Ampère operator
P. Delanoë · 1991
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Habilitation thesis
W. Gangbo · 1995
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Supports of doubly stochastic measures
K. Hestir and S.C. Williams · 1995
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Existence and uniqueness of monotone measure-preserving maps
R.J. McCann · 1995
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Boundary regularity of maps with convex potentials — II
L.A. Caffarelli · 1996
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The geometry of optimal transportation
W. Gangbo and R.J. McCann · 1996
Earlier work this paper cites.
On the design of a reflector antenna
X.-J. Wang · 1996
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A convexity principle for interacting gases
R.J. McCann · 1997
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On the second boundary value problem for equations of Monge-Ampère type
J. Urbas · 1997
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Mass Transportation Problems
S.T. Rachev and L. Rüschendorf · 1998
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A geometrical approach to monotone functions in 𝐑 n \mathbf{R}^{n}
G. Alberti and L. Ambrosio · 1999
Earlier work this paper cites.
Differential equations methods for the Monge-Kantorovich mass transfer problem
L.C. Evans and W. Gangbo · 1999
Cited alongside, same era.
Abstract cyclical monotonicity and Monge solutions for the general Monge-Kantorovich problem
V.L. Levin · 1999
Cited alongside, same era.
Exact solutions to the transportation problem on the line
R.J. McCann · 1999
Cited alongside, same era.
A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem
J.-D. Benamou and Y. Brenier · 2000
Cited alongside, same era.
Shape recognition via Wasserstein distance
W. Gangbo and R.J. McCann · 2000
Cited alongside, same era.
Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality
F. Otto and C. Villani · 2000
Cited alongside, same era.
Hölder regularity of optimal mappings in optimal transportation
J. Liu · 2009
Later among the works it cites.
On the regularity of solutions of optimal transportation problems
G. Loeper · 2009
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Ricci curvature for metric measure spaces via optimal transport
J. Lott and C. Villani · 2009
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Characterization of optimal transport plans for the Monge-Kantorovich problem
W. Schachermayer and J. Teichmann · 2009
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On the second boundary value problem for Monge-Ampère type equations and optimal transportation
N.S. Trudinger and X.-J. Wang · 2009
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Optimal Transport. Old and New
C. Villani · 2009
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A Riemannian interpolation inequality à la Borell, Brascamp and Lieb
D. Cordero-Erausquin, R.J. McCann and M. Schmuckenschläger · 2001
Cited alongside, same era.
Polar factorization of maps on Riemannian manifolds
R.J. McCann · 2001
Cited alongside, same era.
The geometry of dissipative evolution equations: The porous medium equation
F. Otto · 2001
Cited alongside, same era.
On the Monge mass transfer problem
N.S. Trudinger and X.-J. Wang · 2001
Cited alongside, same era.
Constructing optimal maps for Monge’s transport problem as a limit of strictly convex costs
L.A. Caffarelli, M. Feldman and R.J. McCann · 2002
Cited alongside, same era.
Uniqueness and transport density in Monge’s transportation problem
M. Feldman and R.J. McCann · 2002
Cited alongside, same era.
Hedonic price equilibria, stable matching and optimal transport: equivalence, topology and uniqueness
P.-A. Chiappori, R.J. McCann, and L. Nesheim · 2010
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Regularity of optimal transportation maps on compact, locally nearly spherical, manifolds
P. Delanoë and Y. Ge · 2010
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Regularity properties of optimal maps between nonconvex domains in the plane
A. Figalli · 2010
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Partial regularity of Brenier solutions of the Monge-Ampère equation
A. Figalli and Y.-H. Kim · 2010
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Continuity, curvature, and the general covariance of optimal transportation
Y.-H. Kim and R.J. McCann · 2010
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Pseudo-Riemannian geometry calibrates optimal transportation
Y.-H. Kim, R.J. McCann and M. Warren · 2010
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New computable necessary conditions for the regularity theory of optimal transportation
P.W.Y. Lee · 2010
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Interior C 2 , α C^{2,\alpha} regularity for potential functions in optimal transportation
J. Liu, N.S. Trudinger, X.-J. Wang · 2010
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Regularity of optimal transport in curved geometry: the non-focal case
G. Loeper and C. Villani · 2010
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Optimal transportation, topology and uniqueness
N. Ahmad, H.K. Kim, and R.J. McCann · 2011
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The Monge problem in 𝐑 d \mathbf{R}^{d}
T. Champion and L. De Pascale · 2011
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When is multidimensional screening a convex program?
A. Figalli, Y.-H. Kim, and R.J. McCann · 2011
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On the inverse implication of Brenier-McCann theorems and the structure of ( P 2 ( M ) , W 2 ) (P_{2}(M),W_{2})
N. Gigli · 2011
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The Ma-Trudinger-Wang curvature for natural mechanical actions
P.W.Y. Lee and R.J. McCann · 2011
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Regularity of optimal maps on the sphere: the quadratic cost and the reflector antenna
G. Loeper · 2011
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Five lectures on optimal transportation: geometry, regularity, and applications
R.J. McCann and N. Guillen · 2011
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Hölder continuity of optimal multivalued mappings
R.J. McCann and M. Sosio · 2011
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Nearly round spheres look convex
A. Figalli, L. Rifford and C. Villani · 2012
Closest in time.
Rectifiability of optimal transportation plans
R.J. McCann, B. Pass and M. Warren · 2012
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On the local structure of optimal measures in the multi-marginal optimal transportation problem
B. Pass · 2012
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