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

TL;DR

This paper proves that slit carpets cannot be bi-Lipschitz embedded into any uniformly convex Banach space, including Euclidean spaces and L^p spaces, resolving a longstanding open question.

Contribution

It establishes the non-embeddability of slit carpets into uniformly convex Banach spaces, answering a question posed in 1997.

Findings

Slit carpets do not admit bi-Lipschitz embeddings into uniformly convex Banach spaces.

Includes Euclidean spaces and L^p spaces for p in (1,∞).

Resolves Question 8 from Heinonen and Semmes' 1997 list.

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.