Assouad Embedding Theorem

It is easy to show that if a metric space permits a biLipschitz embedding into a Euclidean space then it is doubling. However, the converse of this proposition does not hold. In this talk, I will provide a simple counterexample. The Assouad embedding theorem states that if we perturb the metric of any doubling metric space, then we can always find a biLipschitz embedding into some Euclidean space. I will present the proof of this theorem and provide a few examples in this talk.