TL;DR
This paper introduces a novel metric comparison method related to Alexandrov's comparison, linking it to optimal transport continuity and the MTW condition on Riemannian manifolds.
Contribution
It defines a new metric comparison type and establishes its connection to optimal transport regularity and the MTW condition.
Findings
New metric comparison similar to Alexandrov's
Connection established between the new comparison and optimal transport continuity
Relation to the MTW condition on Riemannian manifolds
Abstract
We define a new type of metric comparison similar to the comparison of Alexandrov. We show that it has strong connections to continuity of optimal transport between regular measures on a Riemannian manifold, in particular to the so called MTW condition introduced by Xi-Nan Ma, Neil Trudinger and Xu-Jia Wang.
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