A characterization of spaces of homogeneous type induced by continuous ellipsoid covers of ${\mathbb{R}}^n$

Abstract

We study the relationship between the concept of a continuous ellipsoid $\Theta$ cover of ${\mathbb{R}}^n$, which was introduced by Dahmen, Dekel, and Petrushev, and the space of homogeneous type induced by $\Theta$. We characterize the class of quasi-distances on ${\mathbb{R}}^n$ (up to equivalence) which correspond to continuous ellipsoid covers. This places firmly continuous ellipsoid covers as a subclass of spaces of homogeneous type on ${\mathbb{R}}^n$ satisfying quasi-convexity and $1$-Ahlfors-regularity.