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Robert J. McCann

  1. Optimal transportation, topology and uniqueness.

    Пт, 08/27/2010 - 15:01 — SomNambul
    Authors: Robert J. McCann, Najma Ahmad, Hwa Kil Kim
    Subjects: Probability
    Abstract

    The Monge-Kantorovich transportation problem involves optimizing with respect
    to a given a cost function. Uniqueness is a fundamental open question about
    which little is known when the cost function is smooth and the landscapes
    containing the goods to be transported possess (non-trivial) topology. This
    question turns out to be closely linked to a delicate problem (# 111) of
    Birkhoff [14]: give a necessary and sufficient condition on the support of a
    joint probability to guarantee extremality among all measures which share its
    marginals.

  2. Rectifiability of Optimal Transportation Plans.

    Чт, 03/25/2010 - 16:01 — SomNambul
    Authors: Robert J. McCann, Brendan Pass, Micah Warren
    Subjects: Analysis of PDEs
    Abstract

    The purpose of this note is to show that the solution to the Kantorovich
    optimal transportation problem is supported on a Lipschitz manifold, provided
    the cost is $C^{2}$ with non-singular mixed second derivative. We use this
    result to provide a simple proof that solutions to Monge's optimal
    transportation problem satisfy a change of variables equation almost
    everywhere.

  3. When is multidimensional screening a convex program?.

    Чт, 12/17/2009 - 16:01 — SomNambul
    Authors: Robert J. McCann, Alessio Figalli, Young-Heon Kim
    Subjects: Optimization and Control
    Abstract

    A principal wishes to transact business with a multidimensional distribution
    of agents whose preferences are known only in the aggregate. Assuming a twist
    (= generalized Spence-Mirrlees single-crossing) hypothesis and that agents can
    choose only pure strategies, we identify a structural condition on the
    preference b(x,y) of agent type x for product type y -- and on the principal's
    costs c(y) -- which is necessary and sufficient for reducing the profit
    maximization problem faced by the principal to a convex program.

  4. Continuity and injectivity of optimal maps for non-negatively cross-curved costs.

    Пнд, 11/23/2009 - 16:01 — SomNambul
    Authors: Robert J. McCann, Alessio Figalli, Young-Heon Kim
    Subjects: Analysis of PDEs
    Abstract

    Consider transportation of one distribution of mass onto another, chosen to
    optimize the total expected cost, where cost per unit mass transported from x
    to y is given by a smooth function c(x,y).

  5. The Ma-Trudinger-Wang curvature for natural mechanical actions.

    Втр, 09/01/2009 - 12:47 — SomNambul
    Authors: Paul W.Y. Lee, Robert J. McCann
    Subjects: Analysis of PDEs
    Abstract

    The Ma-Trudinger-Wang curvature -- or cross-curvature -- is an object arising
    in the regularity theory of optimal transportation. We study this quantity for
    costs defined by natural mechanical actions. We show the least action
    corresponding to a harmonic oscillator has zero cross-curvature, and in
    particular satisfies the necessary and sufficient condition (A3w) for the
    continuity of optimal maps.

  6. The Ma-Trudinger-Wang curvature for natural mechanical actions.

    Втр, 09/01/2009 - 12:47 — SomNambul
    Authors: Paul W.Y. Lee, Robert J. McCann
    Subjects: Analysis of PDEs
    Abstract

    The Ma-Trudinger-Wang curvature -- or cross-curvature -- is an object arising
    in the regularity theory of optimal transportation. We study this quantity for
    costs defined by natural mechanical actions. We show the least action
    corresponding to a harmonic oscillator has zero cross-curvature, and in
    particular satisfies the necessary and sufficient condition (A3w) for the
    continuity of optimal maps.

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