Abstract

Using a result of Y. Brenier [Comm. Pure Appl. Math. 44 (1991) 375–417] we give a representation of the polar cone of monotone gradient fields in terms of measure-preserving maps, or bistochastic measures. Some applications to variational problems subject to a convexity constraint are given.

Author information

Contact details are reproduced from the original publication and may be historical.

Guillaume Carlier

CEREMADE, Université Paris IX Dauphine, Pl. de Lattre de Tassigny, 75775 Paris 16, France
and: UMR CNRS 7534

carlier@ceremade.dauphine.fr

Thomas Lachand-Robert

Lab. de Mathématiques, Université de Savoie, 73376 Le Bourget-du-Lac, France
and: UMR CNRS 5127

Keywords

  • Convexity constraint
  • Euler-Lagrange equation
  • measure-preserving maps
  • bistochastic measures

Suggested citation

G. Carlier, T. Lachand-Robert. “Representation of the Polar Cone of Convex Functions and Applications.” Journal of Convex Analysis 15 (2008), No. 3, 535–546. https://doi.org/10.68381/jca15038

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Published by Heldermann Verlag, 2008. Rights now held by Banach Press.