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In this note we prove estimates for the average cost in the quadratic optimal transport problem on the two-dimensional flat torus which are optimal up to a double logarithm. We also prove sharp estimates on the displacement. This is based on the combination of a post-processing of our quantitative linearization result together with a quasi-orthogonality property.

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Almost Sharp Rates of Convergence for the Average Cost and Displacement in the Optimal Matching Problem

  • 会议论文

  • 发表日期：

    2025年

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  • Michael Goldman,
  • Martin Huesmann,
  • Felix Otto

摘要

In this note we prove estimates for the average cost in the quadratic optimal transport problem on the two-dimensional flat torus which are optimal up to a double logarithm. We also prove sharp estimates on the displacement. This is based on the combination of a post-processing of our quantitative linearization result together with a quasi-orthogonality property.