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 Lipschitz map from Euclidean space to metric must be close factoring through tree. Using results Azzam–Schul and present authors, this gives checkable conditions for have large piece its domain on it behaves like an orthogonal projection. The proof involves new lower bounds continuity statements mapping content, relies “qualitative” version main theorem recently proven by Esmayli–Hajłasz.