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On the nonexistence of bilipschitz parameterizations and geometric problems about $A_\infty$-weights

On the nonexistence of bilipschitz parameterizations and geometric problems about $A_\infty$-weights

How can one recognize when a metric space is bilipschitz equivalent to a Euclidean space? One should not take the abstraction of metric spaces too seriously here; subsets of \mathbb R^n are already quite interesting. It is easy to generate geometric conditions which are necessary for bilipschitz equivalence, but it …