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

TL;DR

This paper characterizes the class of quasi-distances on ^n induced by continuous ellipsoid covers, establishing their place within spaces of homogeneous type with specific regularity properties.

Contribution

It provides a complete characterization of quasi-distances associated with continuous ellipsoid covers and clarifies their relation to spaces of homogeneous type.

Findings

Continuous ellipsoid covers induce specific quasi-distances.

These quasi-distances form a subclass of spaces of homogeneous type.

The characterized spaces satisfy quasi-convexity and 1-Ahlfors-regularity.

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.