A Simple Sampling Method for Metric Measure Spaces

TL;DR

This paper presents a new simple sampling technique for metric measure spaces using a snowflakeing operator, demonstrating its equivalence to sampling subsets of Euclidean space for doubling metric spaces and comparing it with existing methods.

Contribution

Introduces a novel, straightforward sampling method for metric measure spaces based on the snowflakeing operator, linking it to classical results and existing approaches.

Findings

Sampling of doubling metric spaces is bilipschitz equivalent to subsets of Euclidean space.

The new method is compared favorably with existing triangulation approaches.

The approach simplifies sampling in metric measure spaces with curvature conditions.

Abstract

We introduce a new, simple metric method of sampling metric measure spaces, based on a well-known "snowflakeing operator" and we show that, as a consequence of a classical result of Assouad, the sampling of doubling metric spaces is bilipschitz equivalent to that of subsets of some $\mathbb{R}^N$. Moreover, we compare this new method with two other approaches, in particular to one that represents a direct application of our triangulation method of metric measure spaces satisfying a generalized Ricci curvature condition.