Generalized extremizers for t-intersecting families over [0,1]^n

Determine whether, for a product distribution D^n on [0,1]^n with E[D] = p < 1/2 and a fixed threshold t ∈ R, every maximal t-intersecting family A ⊆ [0,1]^n (i.e., ⟨x,y⟩ ≥ t for all x,y ∈ A) is a halfspace of the form A(z,α) = {x ∈ [0,1]^n : ⟨x, z⟩ ≥ α} for some z ∈ [0,1]^n and α ∈ R.

Background

This generalizes the Ahlswede–Khachatrian theorem on maximal t-intersecting families of k-subsets to continuous cubes with arbitrary product distributions. The classical extremal families have a combinatorial ‘initial segment’ structure; here the conjectured extremizers are affine halfspaces in [0,1]^n.

A μp analogue and robustness questions are also discussed, and there are partial analytic results in special cases (e.g., Friedgut’s proof for μp, t=2, and small p).