Regular mappings and non-existence of bi-Lipschitz embeddings for slit carpets

Abstract

We prove that the "slit carpet" introduced by Merenkov does not admit a bi-Lipschitz embedding into any uniformly convex Banach space. In particular, this includes any Euclidean space $\mathbb{R}^n$, but also spaces such as $L^p$ for $p \in (1,\infty)$. This resolves Question 8 in the 1997 list by Heinonen and Semmes.