Calculus of Variations and Geometric Measure Theory
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E. Le Donne - T. Rajala

Assouad dimension, Nagata dimension, and uniformly close metric tangents

created by ledonne on 21 Dec 2018



Inserted: 21 dec 2018

Year: 2013

ArXiv: 1306.5859 PDF


We study the Assouad dimension and the Nagata dimension of metric spaces. As a general result, we prove that the Nagata dimension of a metric space is always bounded from above by the Assouad dimension. Most of the paper is devoted to the study of when these metric dimensions of a metric space are locally given by the dimensions of its metric tangents. Having uniformly close tangents is not sufficient. What is needed in addition is either that the tangents have dimension with uniform constants independent from the point and the tangent, or that the tangents are unique. We will apply our results to equiregular subRiemannian manifolds and show that locally their Nagata dimension equals the topological dimension.

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