A quantitative metric differentiation theorem

  title={A quantitative metric differentiation theorem},
  author={Jonas Azzam and Raanan Schul},
  journal={arXiv: Metric Geometry},
The purpose of this note is to point out a simple consequence of some earlier work of the authors, "Hard Sard: Quantitative implicit function and extension theorems for Lipschitz maps". For $f$, a Lipschitz function from a Euclidean space into a metric space, we give quantitative estimates for how often the pullback of the metric under $f$ is approximately a seminorm. This is a quantitative version of Kirchheim's metric differentiation result from 1994. Our result is in the form of a Carleson… Expand
Quantitative decompositions of Lipschitz mappings into metric spaces
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 whichExpand
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 throughExpand
Coarse differentiation and quantitative nonembeddability for Carnot groups
We give lower bound estimates for the macroscopic scale of coarse differentiability of Lipschitz maps from a Carnot group with the Carnot-Carath\'{e}odory metric $(G,\dcc)$ to a few different classesExpand
Heat flow and quantitative differentiation
For every Banach space $(Y,\|\cdot\|_Y)$ that admits an equivalent uniformly convex norm we prove that there exists $c=c(Y)\in (0,\infty)$ with the following property. Suppose that $n\in \mathbb{N}$Expand
Quantitative affine approximation for UMD targets
It is shown here that if $(Y,\|\cdot\|_Y)$ is a Banach space in which martingale differences are unconditional (a UMD Banach space) then there exists $c=c(Y)\in (0,\infty)$ with the followingExpand
Bi-Lipschitz parts of quasisymmetric mappings
A natural quantity that measures how well a map $f:\mathbb{R}^{d}\rightarrow \mathbb{R}^{D}$ is approximated by an affine transformation isExpand


Bi-Lipschitz Decomposition of Lipschitz functions into a Metric space
We prove a quantitative version of the following statement. Given a Lipschitz function f from the k-dimensional unit cube into a general metric space, one can decomposed f into a finite number ofExpand
Affine Approximation of Lipschitz Functions and Nonlinear Quotients
Abstract. New concepts related to approximating a Lipschitz function between Banach spaces by affine functions are introduced. Results which clarify when such approximations are possible are provedExpand
Rectifiable metric spaces: local structure and regularity of the Hausdorff measure
We consider the question whether the "nice" density behaviour of Hausdorff measure on rectifiable subsets of Euclidian spaces preserves also in the general metric case. For this purpose we show theExpand
The notions of Lipschitz and bilipschitz mappings provide classes of mappings connected to the geometry of metric spaces in certain ways. A notion between these two is given by "regular mappings"Expand
A characterization of potential spaces
A mean oscillation characterization, valid for all a > 0, of the spaces L' of Bessel potentials of LP functions is given and is used to relate the known characterizations for 0 < a < 2 viaExpand
Analysis of and on uniformly rectifiable sets
The notion of uniform rectifiability of sets (in a Euclidean space), which emerged only recently, can be viewed in several different ways. It can be viewed as a quantitative and scale-invariantExpand
Discretization and affine approximation in high dimensions
Lower estimates are obtained for the macroscopic scale of affine approximability of vector-valued Lipschitz functions on finite-dimensional normed spaces, completing the work of Bates, Johnson,Expand
Lectures on discrete geometry
  • J. Matousek
  • Computer Science, Mathematics
  • Graduate texts in mathematics
  • 2002
This book is primarily a textbook introduction to various areas of discrete geometry, in which several key results and methods are explained, in an accessible and concrete manner, in each area. Expand
Rectifiable sets and the Traveling Salesman Problem
Let K c C be a bounded set. In this paper we shall give a simple necessary and sufficient condit ion for K to lie in a rectifiable curve. We say that a set is a rectifiable curve if it is the imageExpand
Bounded Analytic Functions
Preliminaries.- Hp Spaces.- Conjugate Functions.- Some Extremal Problems.- Some Uniform Algebra.- Bounded Mean Oscillation.- Interpolating Sequences.- The Corona Construction.- Douglas Algebras.-Expand