A class of multi-marginal c-cyclically monotone sets with explicit c-splitting potentials

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:461

Article

Bartz, Sedi¹  Bauschke, Heinz H.²  Wang, Xianfu² 

[1] Univ Massachusetts, Math, Lowell, MA 01854 USA

[2] Univ British Columbia, Math, Kelowna, BC V1V 1V7, Canada

关键词: c-Convex;    c-Splitting set;    Cyclically monotone;    Monge-Kantorovich;    Multi-marginal;    Optimal transport;   

DOI  :  10.1016/j.jmaa.2018.01.015

来源: Elsevier

【 摘 要 】

Multi-marginal optimal transport plans are concentrated on c-splitting sets. It is known that, similar to the two-marginal case, c-splitting sets are c-cyclically monotone. Within a suitable framework, the converse implication was very recently established by Griessler. However, for an arbitrary cost c, given a multi-marginal c-cyclically monotone set, the question whether there exists an analogous explicit construction to the one from the two-marginal case of c-splitting potentials is still open. When the margins are one-dimensional and the cost belongs to a certain class, Carlier proved that the two-marginal projections of a c-splitting set are monotone. For arbitrary products of sets equipped with cost functions which are sums of two marginal costs, we show that the two-marginal monotonicity condition is a sufficient condition which does give rise to an explicit construction of o-splitting potentials. Our condition is, in principle, easier to verify than the one of multi-marginal c-cyclic monotonicity. Various examples illustrate our results. We show that, in general, our condition is sufficient; however, it is not necessary. On the other hand, we conclude that when the margins are one-dimensional equipped with classical cost functions, our condition is a characterization of c-spfitting sets and extends classical convex analysis. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

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