Jonas Azzam

University of Edinburgh

Absolute continuity and rectifiability of measures via Wasserstein distance

Analysis and Geometry Seminar

27th February 2018, 3:00 pm – 4:00 pm
Howard House, Second floor seminar room

Given a measure $\mu$ in Euclidean space, we seek conditions that characterize when it is d-rectifiabile, which means it is absolutely continuous to surface measure restricted to a countable union of d-dimensional surfaces. One of the more famous such characterizations is that of David Preiss which says a measure is rectifiable if and only if, $\mu$-a.e., we have $\mu(B(x,r))r^{-d}$ converges to a nonzero constant as r goes to zero. In this talk, we describe a new characterization in terms of a Dini-type condition that quantifies how much a measure deviates from looking like planar Lebesgue measure over all balls centered at a point. By "looks like," we mean with respect to a quantity based on the 1-Wasserstein distance. Roughly speaking, if the distance from $\mu$ restricted to B(x,r) to a multiple of planar Lebesgue measure is square integrable in r, then the measure is rectifiable (the converse was already shown by Tolsa). This is a joint work with Xavier Tolsa and Tatiana Toro.

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