Full characterization of optimal transport plans for concave costs

Paul  Pegon; Filippo Santambrogio; Davide  Piazzoli

doi:10.3934/dcds.2015.35.6113

Article Contents

2015, Volume 35, Issue 12: 6113-6132. Doi: 10.3934/dcds.2015.35.6113

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Full characterization of optimal transport plans for concave costs

1.
Laboratoire de Mathématiques d'Orsay, Université Paris-Sud, 91405 Orsay cedex
2.
Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, 91405 Orsay cedex
3.
Cambridge Centre for Analysis, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WB

Received: October 31, 2013

Published: November 2015

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Abstract

This paper slightly improves a classical result by Gangbo and McCann (1996) about the structure of optimal transport plans for costs that are strictly concave and increasing functions of the Euclidean distance. Since the main difficulty for proving the existence of an optimal map comes from the possible singularity of the cost at $0$, everything is quite easy if the supports of the two measures are disjoint; Gangbo and McCann proved the result under the assumption $\mu(supp(\mathbf{v}))=0$; in this paper we replace this assumption with the fact that the two measures are singular to each other. In this case it is possible to prove the existence of an optimal transport map, provided the starting measure $\mu$ does not give mass to small sets (i.e. $(d\!-\!1)-$rectifiable sets). When the measures are not singular the optimal transport plan decomposes into two parts, one concentrated on the diagonal and the other being a transport map between mutually singular measures.

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Mathematics Subject Classification: 49J45, 49K21, 49Q20, 28A75.

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