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Rectifiability of Optimal Transportation Plans

Published online by Cambridge University Press:  20 November 2018

Robert J. McCann ,
Brendan Pass  and
Micah Warren
Robert J. McCann
Affiliation:
Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4 email: mccann@math.toronto.edu
Brendan Pass
Affiliation:
Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Current address: Department ofMathematical and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1 email: bpass@math.utoronto.capass@ualberta.ca
Micah Warren
Affiliation:
Department of Mathematics, Princeton University, Princeton, NJ, USA 08544 email: mww@princeton.edu
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Abstract

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The regularity of solutions to optimal transportation problems has become a hot topic in current research. It is well known by now that the optimal measure may not be concentrated on the graph of a continuous mapping unless both the transportation cost and the masses transported satisfy very restrictive hypotheses (including sign conditions on the mixed fourth-order derivatives of the cost function). The purpose of this note is to show that in spite of this, the optimal measure is supported on a Lipschitz manifold, provided only that 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.

Keywords

49K2049K6035J9658C07
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 64 , Issue 4 , 01 August 2012 , pp. 924 - 934
Copyright
Copyright © Canadian Mathematical Society 2012

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