A glimpse into the differential topology and geometry of optimal transport

Robert J. McCann

doi:10.3934/dcds.2014.34.1605

Article Contents

2014, Volume 34, Issue 4: 1605-1621. Doi: 10.3934/dcds.2014.34.1605

This issue Previous Article Congestion-driven dendritic growth Next Article Multi-marginal optimal transport and multi-agent matching problems: Uniqueness and structure of solutions

A glimpse into the differential topology and geometry of optimal transport

Robert J. McCann^1,

1.
Department of Mathematics, University of Toronto, Toronto ON Canada M5R 2Y4

Received: June 30, 2012

Revised: October 31, 2012

Published: March 2014

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Abstract

This note exposes the differential topology and geometry underlying some of the basic phenomena of optimal transportation. It surveys basic questions concerning Monge maps and Kantorovich measures: existence and regularity of the former, uniqueness of the latter, and estimates for the dimension of its support, as well as the associated linear programming duality. It shows the answers to these questions concern the differential geometry and topology of the chosen transportation cost. It also establishes new connections --- some heuristic and others rigorous --- based on the properties of the cross-difference of this cost, and its Taylor expansion at the diagonal.

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Mathematics Subject Classification: Primary: 49N60, 58E30; Secondary: 53C50, 90B06, 90C08, 91B32.

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