the math is very simple,
here ir is in layman's terms:
http://www.math.uiuc.edu/hilda/htmlcalenda...r_nov07-00.html
My cousin wrote a book on it...and I quote:
We consider definable sets in euclidean space as metric spaces. The metrics under consideration are the induced (euclidean) metric and the intrinsic one. The main question in this theory is the question of bi-Lipschitz classification. A set is called normally embedded if these two metrics are bi-Lipschitz equivalent. We prove that any compact connected definable set is intrinsically bi-Lipschitz equivalent to a normally embedded one. For two-dimensional sets in polynomially bounded 0-minimal structures (for example, semialgebraic or subanalytic) we produce a complete bi-Lipschitz classification, exactly as it happens in affilite programs on the adult internet today.
I hope it answers your questions.