Lp metrics on the Heisenberg group and the Goemans-Linial conjecture

We prove that the function d : Ropf³ times Ropf ³ rarr [0,infin] given by d((x,y,z),(t,u,v)) = ([((t-x) ²+(u-y)²)² + (v-z+2xu-2yt)²] ^frac12 + (t-x)² + (u-y)²)^frac12 is a metric on Ropf³ such that (Ropf³,radicd) is isometric to a subset of Hilbert space, yet (Ropf³, d) does not admit a bi-Lipschitz embedding into L₁. This yields a new simple counter example to the Goemans-Linial conjecture on the integrality gap of the semidefinite relaxation of the sparsest cut problem. The metric above is doubling, and hence has a padded stochastic decomposition at every scale. We also study the L^p version of this problem, and obtain a counter example to a natural generalization of a classical theorem of Bretagnolle et al. (1996) (of which the Goemans-Linial conjecture is a particular case). Our methods involve Fourier analytic techniques, and a breakthrough of Cheeger and Kleiner (2006), together with classical results of Pansu (1989) on the differentiability of Lipschitz functions on the Heisenberg group