# Duality for rectified Cost Functions

created by pratelli on 25 Sep 2010
modified on 16 Feb 2015

[BibTeX]

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.