Wasserstein Distance and the Rectifiability of Doubling Measures: Part II
Abstract
We study the structure of the support of a doubling measure by analyzing its self-similarity properties, which we estimate using a variant of the L1 Wasserstein distance. We show that measure satisfying certain self-similarity conditions admits a unique (up to multiplication by a constant) flat tangent measure at almost every point. This allows us to decompose the support into rectifiable pieces of various dimensions.
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