A quantitative metric differentiation theorem

Abstract

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 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 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 packing condition.