Lower bounds on mapping content and quantitative factorization through trees

We give a simple quantitative condition, involving the “mapping content” of Azzam–Schul, which implies that a Lipschitz map from a Euclidean space to a metric space must be close to factoring through a tree. Using results of Azzam–Schul and the present authors, this gives simple checkable conditions...

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Published in:Journal of the London Mathematical Society 2022-09, Vol.106 (2), p.1170-1188
Main Authors: David, Guy C., Schul, Raanan
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Language:English
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description We give a simple quantitative condition, involving the “mapping content” of Azzam–Schul, which implies that a Lipschitz map from a Euclidean space to a metric space must be close to factoring through a tree. Using results of Azzam–Schul and the present authors, this gives simple checkable conditions for a Lipschitz map to have a large piece of its domain on which it behaves like an orthogonal projection. The proof involves new lower bounds and continuity statements for mapping content, and relies on a “qualitative” version of the main theorem recently proven by Esmayli–Hajłasz.
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title Lower bounds on mapping content and quantitative factorization through trees
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