Sharp constant for the L1 Poincaré inequality on the Hamming cube for real-valued functions

Determine the exact optimal dimension-free constant C such that for every integer n ≥ 1 and every real-valued function f: {0,1}^n → ℝ, the inequality E[|∇f|] ≥ C · E[|f − E f|] holds, where |∇f(x)| = sqrt(∑_{i=1}^n (1/2 (f(x) − f(x ⊕ e_i)))^2) and E denotes the uniform average over {0,1}^n.

Background

The paper proves the sharp L1 Poincaré inequality with constant 1 for Boolean-valued functions f: {0,1}n → {0,1}, showing ||∇f||_1 ≥ ||f − E f||_1. For general real-valued functions on the cube, the best dimension-free constant in the analogous inequality remains undetermined.

Existing results place this constant in the interval (2/π, √(2/π)], with improvements appearing in the literature cited by the authors. Establishing the exact value would close a longstanding endpoint problem for L1 Poincaré-type inequalities on the hypercube.

References

Note that it is still open to determine the sharp constant in this inequality for {\it real-valued} f, but it is known that this constant lies in the interval $(\frac2{\pi},\sqrt{\frac{2}{\pi}]$; see , , also , for quantum and vector-valued inequalities.

— Sharp isoperimetric inequalities on the Hamming cube II: The critical exponent  (2602.20462 - Durcik et al., 24 Feb 2026) in Introduction, Applications section, after Theorem 1.2