Hellinger conjecture for balanced Boolean functions under noise

Establish that for every integer n ≥ 1, every balanced Boolean function f: {0,1}^n → {±1} with E f = 0, and every correlation parameter ρ ∈ [−1,1], the inequality E[√(1 − (T_ρ f)^2)] ≥ √(1 − ρ^2) holds, where T_ρ is the standard noise operator on the Boolean cube (mapping f(x) to E_Z f(x ⊕ Z) with independent bits Z_j flipped with probability (1 − ρ)/2).

Background

The authors relate their isoperimetric results to conjectures in information theory about noisy Boolean channels. The Hellinger conjecture implies the Courtade–Kumar (most informative Boolean function) conjecture and predicts a specific inequality involving the noise operator T_ρ.

While the paper confirms the low-noise limit of this prediction for balanced functions via their main isoperimetric result, the full Hellinger conjecture remains unproven in general, motivating its explicit restatement.

References

In this case the Hellinger conjecture asserts that for every correlation parameter $\rho \in [-1,1]$, \begin{equation}\label{eqn:hellinger} \mathbf{E}\sqrt{1 - (T_\rho f)2} \ge \sqrt{1-\rho2}, \end{equation} where $T_\rho f$ is the standard noise operator (see ).

eqn:hellinger:

E1−(Tρf)2≥1−ρ2,\mathbf{E}\sqrt{1 - (T_\rho f)^2} \ge \sqrt{1-\rho^2},

— Sharp isoperimetric inequalities on the Hamming cube II: The critical exponent  (2602.20462 - Durcik et al., 24 Feb 2026) in Section 6 (Connection to the Hellinger conjecture)