JournalsrmiVol. 28, No. 4pp. 1123–1142

Assouad’s theorem with dimension independent of the snowflaking

  • Assaf Naor

    New York University, United States
  • Ofer Neiman

    Ben Gurion University of the Negev, Beer Sheva, Israel
Assouad’s theorem with dimension independent of the snowflaking cover
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Abstract

It is shown that for every K>0K>0 and ε(0,1/2)\varepsilon\in (0,1/2) there exist N=N(K)NN=N(K)\in \mathbb{N} and D=D(K,ε)(1,)D=D(K,\varepsilon)\in (1,\infty) with the following properties. For every metric space (X,d)(X,d) with doubling constant at most KK, the metric space (X,d1ε)(X,d^{1-\varepsilon}) admits a bi-Lipschitz embedding into RN\mathbb{R}^N with distortion at most DD. The classical Assouad embedding theorem makes the same assertion, but with NN\to \infty as ε0\varepsilon\to 0.

Cite this article

Assaf Naor, Ofer Neiman, Assouad’s theorem with dimension independent of the snowflaking. Rev. Mat. Iberoam. 28 (2012), no. 4, pp. 1123–1142

DOI 10.4171/RMI/706