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$\ell_p$ (p>2) does not coarsely embed into a Hilbert space

W. B. Johnson·N. L. Randrianarivony
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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: ϕ_1(d(x,y)) \leq d(f(x),f(y)) \leq ϕ_2(d(x,y)) where ϕ_1 and ϕ_2 are nondecreasing functions on [0,\infty) with values in [0,\infty), with the condition that ϕ_1(t) tends to \infty as t tends to \infty. We show that \ell_p does not coarsely embed in a Hilbert space for 2<p<\infty.

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W. B. Johnson, N. L. Randrianarivony. 2004-10-19. $\ell_p$ (p>2) does not coarsely embed into a Hilbert space. https://arxiv.org/abs/math/0410427

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