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

James R. Lee, Assaf Naor

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

We prove that the function d : R3 x R3 → [0, ∞) given by d((x, y, z), (t, u, v)) = ( [((i - x)2 + (u - y)2f +(v - z + 2xu - 2yt)2]1/2 + (t - x) 2 + (u - y)2)1/2. is a metric on ℝ3 such that (ℝ3, √d) is isometric to a subset of Hubert space, yet (R3, 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 Lp version of this problem, and obtain a counter example to a natural generalization of a classical theorem of Bretagnolle, Dacunha-Castelle and Krivine (of which the Goemans-Linial conjecture is a particular case). Our methods involve Fourier analytic techniques, and a recent breakthrough of Cheeger and Kleiner, together with classical results of Pansu on the differentiability of Lipschitz functions on the Heisenberg group.

Original languageEnglish (US)
Title of host publication47th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2006
Pages99-108
Number of pages10
DOIs
Publication statusPublished - 2006
Event47th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2006 - Berkeley, CA, United States
Duration: Oct 21 2006Oct 24 2006

Other

Other47th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2006
CountryUnited States
CityBerkeley, CA
Period10/21/0610/24/06

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ASJC Scopus subject areas

  • Engineering(all)

Cite this

Lee, J. R., & Naor, A. (2006). Lp metrics on the Heisenberg group and the Goemans-Linial conjecture. In 47th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2006 (pp. 99-108). [4031347] https://doi.org/10.1109/FOCS.2006.47