Sharp multivariate convex sub-Gaussian comparison constant (d ≥ 2)

Determine the smallest constant c > 0, independent of the dimension d, such that for every integer d ≥ 2 and every integrable random vector X ∈ R^d with E[X] = 0 and directional sub-Gaussian tails P(|⟨v, X⟩| > t) ≤ 2e^{-t^2/2} for all unit vectors v and all t ≥ 0, the convex domination inequality E[f(X)] ≤ E[f(c G)] holds for every convex function f : R^d → R for which the expectations are finite, where G ∼ N(0, I_d) is the standard Gaussian vector.

Background

A theorem of van Handel guarantees the existence of a dimension-free constant c such that any random vector X in R^d with directional sub-Gaussian tails is dominated in convex order by c times a standard Gaussian vector. While this note determines the exact sharp constant in dimension one and provides analogous one-dimensional results under sub-exponential tails, the general multivariate setting remains unresolved.

The authors establish the exact one-dimensional constant c_⋆ and construct an extremal distribution achieving it. They also derive higher-dimensional consequences in structured settings (a sequential tensorization principle and a comparator for convex ridge functions). However, obtaining the optimal universal constant for full multivariate convex domination under the directional sub-Gaussian tail bound is still an open problem.