Abstract
It is well-known that duality in the Monge–Kantorovich transport problem holds true provided that the cost function c : X × Y → [0, ∞] is lower semi-continuous or finitely valued, but it may fail otherwise. We present a suitable notion of rectification c r of the cost c, so that the Monge-Kantorovich duality holds true replacing c by c r . In particular, passing from c to c r only changes the value of the primal Monge–Kantorovich problem. Finally, the rectified function c r is lower semi-continuous as soon as X and Y are endowed with proper topologies, thus emphasizing the role of lower semi-continuity in the duality-theory of optimal transport.
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Communicated by N. Trudinger.
The first author acknowledges financial support from the Austrian Science Fund (FWF) under Grant P21209. The work of the second author was partially supported by the ERC Starting Grant Analysis of optimal sets and optimal constants: old questions and new results and by the MEC of Spain Government through the 2008 project MTM2008-03541. The second author gratefully acknowledges the hospitality of the University of Vienna.