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

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.