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We study the quantitative properties of Lipschitz mappings from Euclidean spaces into metric spaces. We prove that it is always possible to decompose the domain of such a mapping into pieces on which the mapping “behaves like a projection mapping” along with a “garbage set” that isarbitrarily smallin an appropriate sense. Moreover, our control is quantitative, i.e., independent of both the particular mapping and the metric space it maps into. This improves a theorem of Azzam-Schul from the paper “Hard Sard”, and answers a question left open in that paper. The proof uses ideas of quantitative differentiation, as well as a detailed study of how to supplement Lipschitz mappings by additional coordinates to form bi-Lipschitz mappings.more » « less
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A quasiconformal tree is a doubling metric tree in which the diameter of each arc is bounded above by a fixed multiple of the distance between its endpoints. In this paper we show that every quasiconformal tree bi-Lipschitz embeds in some Euclidean space, with the ambient dimension and the bi-Lipschitz constant depending only on the doubling and bounded turning constants of the tree. This answers Question 1.6 of David and Vellis [Illinois J. Math. 66 (2022), pp. 189–244].more » « less
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We give a simple quantitative condition, involving the “mapping content” of Azzam–Schul, which 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.more » « less
