Shortcut Laakso spaces, pure PI unrectifiability and differentiability of Lipschitz functions

TL;DR

This paper constructs new Lipschitz differentiability spaces using shortcut Laakso spaces, explores their geometric properties, and characterizes when these spaces are PI rectifiable or Y-LDS, advancing understanding of differentiability in metric spaces.

Contribution

It introduces a family of purely PI unrectifiable Lipschitz differentiability spaces based on shortcut Laakso spaces and characterizes their rectifiability and differentiability properties.

Findings

Laakso spaces satisfy the necessary hypotheses for Lipschitz differentiability.

The paper characterizes when shortcut spaces are PI rectifiable or Y-LDS.

Constructs examples of unrectifiable Lipschitz differentiability spaces.

Abstract

We construct a family of purely PI unrectifiable Lipschitz differentiability spaces and investigate the possible of Banach spaces targets for which Lipschitz differentiability holds. We provide a general investigation into the geometry of \emph{shortcut} metric spaces and characterise when such spaces are PI rectifiable, and when they are $Y$-LDS, for a given $Y$. The family of spaces arises as an example of our characterisations. Indeed, we show that Laakso spaces satisfy the required hypotheses.