An example of a differentiability space which is PI-unrectifiable
Metric Geometry
2016-11-08 v1 Classical Analysis and ODEs
Abstract
We construct a (Lipschitz) differentiability space which has at generic points a disconnected tangent and thus does not contain positive measure subsets isometric to positive measure subsets of spaces admitting a Poincar\'e inequality. We also prove that -valued Lipschitz maps are differentiable a.e., but there are also Lipschitz maps taking values in some other Banach spaces having the Radon-Nikodym property which fail to be differentiable on sets of positive measure.
Cite
@article{arxiv.1611.01615,
title = {An example of a differentiability space which is PI-unrectifiable},
author = {Andrea Schioppa},
journal= {arXiv preprint arXiv:1611.01615},
year = {2016}
}