Talagrand’s boundary–influence conjecture

Establish the inequality E_{x∈{−1,1}^n}[√{h_f(x)}] ≥ β · Var(f) · √{log(e / ∑_{i=1}^n Inf_i(f)^2)} for all Boolean functions f:{−1,1}^n→{−1,1}, where h_f(x) is the sensitivity of f at x (the number of hypercube edges at x along which f changes value), Inf_i(f) is the influence of coordinate i under the uniform measure on {−1,1}^n, and β>0 is a universal constant.

Background

The paper surveys classical inequalities relating variance and influences of Boolean functions, including the Poincaré and KKL inequalities, and Talagrand’s sensitivity-based inequality. Motivated by the gap between surface-area notions based on total influence and on expected sensitivity, Talagrand proposed strengthening his sensitivity bound with a logarithmic factor involving the squared influences.

The conjecture posits a universal constant β>0 such that the expected square-root sensitivity lower-bounds variance times a logarithmic term of the inverse of the sum of squared influences. This would recover the KKL logarithmic improvement in regimes where variance is Ω(1) and connect different boundary notions. The present paper later proves this conjecture and even strengthens it, but the conjecture is stated explicitly in the background as the motivating open claim.