*Published Paper*

**Inserted:** 25 sep 2010

**Last Updated:** 16 feb 2015

**Journal:** Calc. Var. PDE

**Year:** 2012

**Abstract:**

It is well-known that duality in the Monge-Kantorovich transport problem holds true provided that the cost function $ c:X \times Y\rightarrow [0,\infty ] $ 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.

**Keywords:**
Mass transportation, Monge-Kantorovich duality

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