Lower bounds on mapping content and quantitative factorization through trees

TL;DR

This paper establishes conditions under which Lipschitz maps from Euclidean spaces to metric spaces are close to factoring through a tree, using new bounds on mapping content and continuity properties.

Contribution

It introduces a simple quantitative condition involving mapping content that ensures Lipschitz maps approximate orthogonal projections on large subsets.

Findings

Provides new lower bounds for mapping content.

Establishes continuity properties for mapping content.

Offers checkable conditions for Lipschitz maps to behave like projections.

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--Haj{\l}asz.