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

Abstract

We show that, for all $\alpha \ge 0$, 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.

19 pages, 3 figures, minor changes on the original version

References in corpus (1)

• Quasisymmetric spheres over Jordan domains