Lower bounds on mapping content and quantitative factorization through trees

2021/07/02 by David, Guy C., Schul, Raanan
#2010 MSC: 28A75 #30L99 #53C23 #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2107.01108

Abstract

We give a simple quantitative condition, involving the "mapping content" of Azzam--Schul, that 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.

Related

• Lipschitz mappings, metric differentiability, and factorization through metric trees
• Quantitative decompositions of Lipschitz mappings into metric spaces
• Uniformly continuous maps between ends of R-trees
• A Note on the Lipschitz Selection
• Matchings in metric spaces, the dual problem and calibrations modulo 2