Talagrand’s conjecture on boundary-based isoperimetric inequality for the p-biased hypercube

Determine whether for every Boolean function f: {−1,1}^n → {−1,1} on the p-biased discrete hypercube with product measure μ_p, the boundary functional h_f defined by h_f(x) equal to the number of edges from x to A^c when x ∈ A = {y : f(y)=1} and 0 otherwise satisfies the inequality E[h_f] ≥ C_p Var(f) log(1 + e / Σ_{j=1}^n (Inf_j(f))^2), where C_p > 0 depends only on p and Inf_j(f) = E_{μ_p} |f − E_{x_j∼μ_p} f| denotes the L^1 influence with respect to coordinate j.

Background

The paper discusses the Eldan–Gross inequality on the (possibly biased) discrete hypercube and in more general Markov diffusion settings. The Eldan–Gross bound involves the discrete gradient |∇f|, whereas Talagrand’s 1997 conjecture concerns the boundary-counting functional h_f on the p-biased hypercube.

The authors recall Talagrand’s conjecture as motivation and note that, while |∇f| ≥ h_f pointwise, the two quantities are not generally comparable in expectation. They prove a gradient-based inequality in the p-biased case (Theorem 3), but this does not resolve the conjecture stated with h_f. The conjecture, if true, would imply the Eldan–Gross inequality.