Volume 8 (2025)
De Philippis, Guido;  Shenfeld, Yair
Optimal transport maps, majorization, and log-subharmonic measures
Annales Henri Lebesgue, Volume 8 (2025), pp. 925-963

Metadata

DOI10.5802/ahl.251
Keywords optimal transport ,  majorization ,  Log-subharmonic measures

Abstract

Caffarelli’s contraction theorem bounds the derivative of the optimal transport map between a log-convex measure and a strongly log-concave measure. We show that an analogous phenomenon holds on the level of the trace: the trace of the derivative of the optimal transport map between a log-subharmonic measure and a strongly log-concave measure is bounded. We show that this trace bound has a number of consequences pertaining to volume-contracting transport maps, majorization and its monotonicity along Wasserstein geodesics, growth estimates of log-subharmonic functions, the Wehrl conjecture for Glauber states, and two-dimensional Coulomb gases. We also discuss volume-contraction properties for the Kim–Milman transport map.


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