Duality for rectified cost functions

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.