The L¹ Poincaré constant of the Hamming cube

Let C be the smallest constant such that E|f − Ef| ≤ C E|∇f| for every real-valued function f on every Hamming cube {−1,1}ⁿ, where |∇f| is the Euclidean length of (D₁f, …, D_nf) and D_jf(x) = (f(x) − f(x^(j)))/2. Ben Efraim and Lust-Piquard proved C ≤ π/2, the Gaussian case gives C ≥ √(π/2), and Ivanisvili, Li, van Handel and Volberg asked whether C = √(π/2). We prove that it is, with an explicit constant c_n < √(π/2) in dimension n. The proof turns on an integral of first absolute moments of sums of independent biased random signs: we compute it exactly when the coefficients are nonnegative, and show, with a sampled Fejér kernel, that it can only grow when the coefficients are replaced by their absolute values. The equality C = √(π/2) is also proved in Lean 4 with Mathlib.

The programs, the certificate and the Lean projects are archived at doi:10.5281/zenodo.23715901 and at github.com/tejasvi/hamming-cube-poincare-v2/tree/v3.