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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We would like to thank an anonymous referee and Alexander Lytchak for thoughtful and constructive comments to the preliminary version of this paper.
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The first author was partially supported by RFBR Grant 17-01-00128, the second author was partially supported by NSF Grant DMS 1309340, the third author was supported in part by DFG Grant SPP 2026 and RFBR Grant 17-01-00128. Essential part of this work was done during the intense activity period “Metric Measure Spaces and Ricci Curvature” September 4–29, 2017 in Bonn.
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Lebedeva, N., Petrunin, A. & Zolotov, V. Bipolar comparison. Geom. Funct. Anal. 29, 258–282 (2019). https://doi.org/10.1007/s00039-019-00481-9