Lower bounds on mapping content and quantitative factorization through trees

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--Haj{l}asz.