Infinitesimal structure of differentiability spaces, and metric differentiation

TL;DR

This paper proves metric differentiation for Cheeger differentiability spaces, providing new proofs for properties of Lipschitz functions on PI spaces and establishing nonembeddability results.

Contribution

It introduces a new proof of metric differentiation in Cheeger differentiability spaces and derives several important corollaries and nonembeddability results.

Findings

Minimal generalized upper gradient equals pointwise Lipschitz constant on PI spaces

Lip-lip constant of Lip-lip spaces is 1

New nonembeddability results for certain metric spaces

Abstract

We prove metric differentiation for differentiability spaces in the sense of Cheeger. As corollaries we give a new proof that the minimal generalized upper gradient coincides with the pointwise Lipschitz constant for Lipschitz functions on PI spaces, a proof that the Lip-lip constant of any Lip-lip space in the sense of Keith is equal to $1$, and new nonembeddability results.