Characterization of quasi-Banach spaces which coarsely embed into a   Hilbert space
  A map f between two metric spaces (X,d_1) and (Y,d_2) is called a coarse embedding of X into Y if there exist two nondecreasing functions phi_1, phi_2:[0,\infty) --> [0,\infty) such that:   phi_1(d_1(x,y)) \leq d_2(f(x),f(y)) \leq phi_2(d_1(x,y)) for all x, y in X,   and phi_1(t) tends to \infty as t tends to \infty.   We characterize those quasi-Banach spaces that have a coarse embedding into a Hilbert space. 
N. L. Randrianarivony