Vertical Versus Horizontal Poincare Inequalities
summary
Proves a concrete vertical vs horizontal inequality on the discrete Heisenberg group that yields sharp lower bounds: any embedding of the ball of radius n (size ~ n^4) into a q‑uniformly convex Banach space has distortion at least (log n)^{1/q} (for Lp this recovers root log n when p<2 and log^{1/p} n when p>2). Proof uses Littlewood–Paley semigroup methods (Fourier/Paley theory) instead of metric differentiation, tracks vertical jumps of size ~√k, and connects to hardness for the Goemans–Linial sparsest‑cut relaxation.
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