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Title of Journal: Calc Var

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Abbravation: Calculus of Variations and Partial Differential Equations

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Springer-Verlag

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10.1007/bf02940764

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1432-0835

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Duality for rectified cost functions

Authors: Mathias Beiglböck Aldo Pratelli
Publish Date: 2011/08/30
Volume: 45, Issue: 1-2, Pages: 27-41
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Abstract

It is wellknown that duality in the Monge–Kantorovich transport problem holds true provided that the cost function c X × Y → 0 ∞ is lower semicontinuous or finitely valued but it may fail otherwise We present a suitable notion of rectification c r of the cost c so that the MongeKantorovich 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 semicontinuous as soon as X and Y are endowed with proper topologies thus emphasizing the role of lower semicontinuity in the dualitytheory of optimal transportThe 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 MTM200803541 The second author gratefully acknowledges the hospitality of the University of Vienna

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