Assouad's theorem with dimension independent of the snowflaking

2010/12/10 by Naor, Assaf, Neiman, Ofer · 2 citations
#FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1012.2307

Abstract

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

Cited by

• The Heinonen-Semmes problems after thirty years
• Learning from one graph: transductive learning guarantees via the geometry of small random worlds

Related

• A constructive proof of the Assouad embedding theorem with bounds on the dimension
• A non-injective Assouad-type theorem with sharp dimension
• How much can we extend the Assouad embedding theorem?
• Accessible values of Assouad and the lower dimensions of subsets
• On Lipschitz partitions of unity and the Assouad--Nagata dimension