Lower bounds on mapping content and quantitative factorization through trees

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.

Lipschitz mappings, metric differentiability, and factorization through metric trees

“…Recently David and Schul [6] used our result (implication (e) ⇒ (a)) to prove a quantitative part of Conjecture 1.13 from [5] which states that if the content  2,1 ∞ (𝑓, 𝑄 𝑜 ) is small, then 𝑓 is close to a mapping g that factors through a tree.…”