arXiv:2002.02022 Abstract | arXiv Analytics

arXiv:2002.02022 [math.AP]Abstract References Reviews Resources

Quantitative stability in the geometry of semi-discrete optimal transport

Published 2020-02-05Version 1

We show quantitative stability results for the geometric "cells" arising in semi-discrete optimal transport problems. Our results show two types of stability, the first is stability of the associated Laguerre cells in measure, without any connectedness or regularity assumptions on the source measure. The second is stability in Hausdorff measure, under a Poincar{\`e}-Wirtinger inequality and a regularity assumption equivalent to the Ma-Trudinger-Wang conditions of regularity in Monge-Amp{\`e}re. This last result also yields stability in the uniform norm of the dual potential functions, all three stability results come with explicit quantitative bounds. Our methods utilize a combination of graph theory, convex geometry, and Monge-Amp{\`e}re regularity theory.

Comments: 24 pages, part of the submission arXiv:1908.11533 was split into this paper

Categories: math.AP, math.OC

Subjects: 49K40, 35J96

Keywords: quantitative stability, semi-discrete optimal transport problems, dual potential functions, stability results come, regularity assumption equivalent

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