Revista Matemática Iberoamericana

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Volume 28, Issue 4, 2012, pp. 1123–1142
DOI: 10.4171/RMI/706

Published online: 2012-10-14

Assouad’s theorem with dimension independent of the snowflaking

Assaf Naor[1] and Ofer Neiman[2]

(1) New York University, United States
(2) Ben Gurion University of the Negev, Beer Sheva, Israel

It is shown that for every $K>0$ and $\varepsilon\in (0,1/2)$ there exist $N=N(K)\in \mathbb{N}$ and $D=D(K,\varepsilon)\in (1,\infty)$ with the following properties. For every metric space $(X,d)$ with doubling constant at most $K$, the metric space $(X,d^{1-\varepsilon})$ admits a bi-Lipschitz embedding into $\mathbb{R}^N$ with distortion at most $D$. The classical Assouad embedding theorem makes the same assertion, but with $N\to \infty$ as $\varepsilon\to 0$.

Keywords: Doubling metric spaces, Assouad’s theorem

Naor Assaf, Neiman Ofer: Assouad’s theorem with dimension independent of the snowflaking. Rev. Mat. Iberoam. 28 (2012), 1123-1142. doi: 10.4171/RMI/706