Majority Is Least Stable Conjecture

Prove that for any linear threshold function f: {−1,1}^n → {−1,1} with n odd and for all ρ ∈ [0,1], the noise stability satisfies Stabρ[f] ≥ Stabρ[Maj_n].

Background

The conjecture posits majority as the extremal least stable linear threshold function across all levels of noise. It implies precise bounds on noise sensitivity and, in the limit ρ→0, a quantitative lower bound on the first-level Fourier weight for balanced LTFs.

Current best general upper bounds on noise sensitivity for LTFs are due to Peres. The consequence at ρ→0 yields a separate open problem on the sum of squared linear coefficients for balanced LTFs.

References

By taking ρ → 0, the conjecture has the following consequence, which is also open: Let f be a linear threshold function with E[f] = 0. Then ∑_{i=1}n ĥf(i)2 ≥ 2/π.

— Open Problems in Analysis of Boolean Functions  (1204.6447 - O'Donnell, 2012) in Main matter, problem “Majority Is Least Stable Conjecture,” remarks, third bullet