Existence and uniqueness of optimal transport maps for signed measures are proved, preserving Hausdorff dimension and Ahlfors regularity of fractal sets via coupled Monge-Ampere equations and adaptive regularization.
(2009).Optimal Transport: Old and New
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Optimal transport of signed measures: existence, uniqueness and fractal structure
Existence and uniqueness of optimal transport maps for signed measures are proved, preserving Hausdorff dimension and Ahlfors regularity of fractal sets via coupled Monge-Ampere equations and adaptive regularization.