On the equality between Monge's infimum and Kantorovich's minimum in optimal mass transportation

This paper concerns the Monge's transport problem in a general Polish space. We find optimal conditions to establish the equality between the infimum of Monge's problem and the minimum of the Kantorovich's relaxed version of the problem. A preliminary version of the results of this paper is contained in the Ph.D. thesis [A. Pratelli, Existence of optimal transport maps and regularity of the transport density in mass transportation problems, Ph.D. Thesis, Scuola Normale Superiore, Pisa, Italy, 2003. Available on http://cvgmt.sns.it/].

RésuméCe papier considère le problème du transport de Monge dans un espace Polonais général. Nous trouvons des conditions optimales pour établir l'égalité entre l'infimum du problème de Monge et le minimum de la version relâchée du problème de Kantorovich. Une version préliminaire des resultats de ce papier est contenue dans la thèse de Doctorat [A. Pratelli, Existence of optimal transport maps and regularity of the transport density in mass transportation problems, Ph.D. Thesis, Scuola Normale Superiore, Pisa, Italy, 2003. Available on http://cvgmt.sns.it/].