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DCDS

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.

keywords:
mean.
,
Monge-Kantorovich
,
curvature
,
Optimal transport
,
optimal maps
,
Ricci
,
pseudo-Riemannian
,
regularity
,
Monge-Ampère type
,
sectional

## Year of publication

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