The sharp one-dimensional convex sub-Gaussian comparison constant

TL;DR

This paper finds the exact smallest constant for convex domination of sub-Gaussian variables by Gaussian, with explicit formulas and applications to higher-dimensional convex comparison principles.

Contribution

It explicitly determines the sharp convex sub-Gaussian comparison constant and extends the results to multivariate convex domination and Gaussian comparison in higher dimensions.

Findings

The sharp constant c* is approximately 2.30952.

An explicit system of equations characterizes c* and the extremal distribution.

Applications include a tensorization principle and a dimension-free Gaussian comparator.

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).