The sharp one-dimensional convex sub-Gaussian comparison constant

Abstract

Let $X$ be an integrable real random variable with mean zero and two-sided sub-Gaussian tail $\mathbb{P}(|X|>t)\le 2e^{-t^2/2}$ for all $t\ge 0$. We determine the smallest constant $c_{\star }$ such that $X$ is dominated in convex order by $c_{\star }G$, where $G$ is standard normal. Equivalently, $c_{\star }^2$ is the sharp one-dimensional convex sub-Gaussian comparison constant appearing in the \emph{Optimization Constants in Mathematics} repository~\cite{optimization-constants-repo}. We show that $c_{\star }$ is given by an explicit system of one-dimensional equations and is attained by an extremal distribution that saturates the tail constraint. Numerically, $c_{\star }\approx 2.30952$ (so $c_{\star }^2\approx 5.33386$). We also determine the analogous sharp constant under a two-sided sub-exponential tail bound, with convex domination by a scaled Laplace law. Finally, we record two higher-dimensional consequences: a sequential tensorization principle for multivariate convex domination, and a dimension-free Gaussian comparator for the cone generated by convex ridge functions (the linear convex order).