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Journal of Convex Analysis 15 (2008), No. 3, 535--546
Copyright Heldermann Verlag 2008

Representation of the Polar Cone of Convex Functions and Applications

Guillaume Carlier
CEREMADE, Université Paris IX Dauphine, Pl. de Lattre de Tassigny, 75775 Paris 16, France

Thomas Lachand-Robert
Lab. de Mathématiques, Université de Savoie, 73376 Le Bourget-du-Lac, France

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.

Keywords: Convexity constraint, Euler-Lagrange equation, measure-preserving maps, bistochastic measures.

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