ℓ_p (p>2) does not coarsely embed into a Hilbert space

2004/10/19 by W. B. Johnson, Johnson, W. B., N. L. Randrianarivony +1
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #math.FA

paper · pdf · doi:10.48550/arxiv.math/0410427

10 pages

arxiv created 2004/10/19 · arxiv updated 2016/09/07

Abstract

A coarse embedding of a metric space X into a metric space Y is a map f: X-->Y satisfying for every x, y in X: ϕ₁(d(x,y)) ≤ d(f(x),f(y)) ≤ ϕ₂(d(x,y)) where ϕ₁ and ϕ₂ are nondecreasing functions on [0,∞) with values in [0,∞), with the condition that ϕ₁(t) tends to ∞ as t tends to ∞. We show that ℓ_p does not coarsely embed in a Hilbert space for 2<p<∞.

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