Bi-Lipschitz embedding of the generalized Grushin plane in Euclidean spaces

TL;DR

This paper proves that the generalized Grushin plane can be embedded in Euclidean space with a bi-Lipschitz map, extending previous results and establishing the minimal embedding dimension for all non-negative parameters.

Contribution

It establishes a sharp bi-Lipschitz embedding of the generalized Grushin plane into Euclidean space for all non-negative parameters, generalizing Wu's recent findings.

Findings

Bi-Lipschitz homeomorphism for all α ≥ 0

Target dimension is optimal

Generalizes previous results by Wu

Abstract

We show that, for all $\alpha\geq 0$, the generalized Grushin plane $\mathbb{G}_{\alpha}$ is bi-Lipschitz homeomorphic to a $2$-dimensional quasiplane in the Euclidean space $\mathbb{R}^{[\alpha ]+2}$, where $[\alpha]$ is the integer part of $\alpha$. The target dimension is sharp. This generalizes a recent result of Wu.