Conformal grushin spaces

We introduce a class of metrics on Rⁿ generalizing the classical Grushin plane. These are length metrics defined by the line element ds = d_E(., Y )^-βds_E for a closed nonempty subset Y ⊂ Rⁿ and ß ε [0, 1). We prove, assuming a Hölder condition on the metric, that these spaces are quasisymmetrically equivalent to Rⁿ and can be embedded in some larger Euclidean space under a bi-Lipschitz map. Our main tool is an embedding characterization due to Seo, which we strengthen by removing the hypothesis of uniform perfectness. In the two-dimensional case, we give another proof of bi-Lipschitz embeddability based on growth bounds on sectional curvature.