One-dimensional convex sub-Gaussian comparison constant

Description of constant

Let $X$ be an integrable real random variable. We say that $X$ is $1$ -sub-Gaussian in the tail sense if $E [X] = 0 and P (∣ X ∣ > t) \leq 2 e^{- t^{2} /2} for all t \geq 0.$ <a href="#DP2026-abs-solved">[DP2026-abs-solved]</a> <a href="#DP2026-def-tail">[DP2026-def-tail]</a>

For real random variables $X, Y$ , write $X ⪯_{c x} Y$ if $E [f (X)] \leq E [f (Y)]$ for every convex $f : R \to R$ for which both expectations are finite. <a href="#DP2026-def-constant">[DP2026-def-constant]</a>

Let $G \sim N (0, 1)$ be standard normal. We define $C_{48} := in f {C > 0 : \forall 1-sub-Gaussian X, X ⪯_{c x} C G} .$ <a href="#DP2026-def-constant">[DP2026-def-constant]</a>

Davis and Power proved that this one-dimensional problem is solved: if $c_{⋆}$ denotes the sharp comparison factor, then $C_{48} = c_{⋆}^{2} = c_{0}^{2}$ , where $c_{0}$ is determined by an explicit system of one-dimensional equations. Numerically, $C_{48} = c_{0}^{2} \approx 5.33386.$ <a href="#DP2026-abs-solved">[DP2026-abs-solved]</a> <a href="#DP2026-thm1-sharp">[DP2026-thm1-sharp]</a> <a href="#DP2026-rem2-num">[DP2026-rem2-num]</a>

Known upper bounds

Bound	Reference	Comments
$< \infty$	<a href="#vH25">[vH25]</a>	Historical finiteness bound: van Handel proved that a universal Gaussian comparator exists in every dimension. <a href="#vH25-thm11">[vH25-thm11]</a>
$c_{0}^{2} \approx 5.33386$	<a href="#DP2026">[DP2026]</a>	Solves the one-dimensional problem exactly. <a href="#DP2026-thm1-sharp">[DP2026-thm1-sharp]</a> <a href="#DP2026-rem2-num">[DP2026-rem2-num]</a>

Known lower bounds

Bound	Reference	Comments
$1$	Elementary	Taking $X \sim N (0, 1)$ and testing with $f (x) = x^{2}$ gives $1 = E [X^{2}] \leq E [Z^{2}] = C$ , so any admissible $C$ must satisfy $C \geq 1$ .
$c_{0}^{2} \approx 5.33386$	<a href="#DP2026">[DP2026]</a>	The sharpness part of Theorem 1 shows that no smaller constant works. <a href="#DP2026-thm1-sharp">[DP2026-thm1-sharp]</a> <a href="#DP2026-rem2-num">[DP2026-rem2-num]</a>

Additional comments and links

Solved case. The paper of Davis and Power determines the sharp one-dimensional constant and shows that it is attained by an extremal distribution saturating the tail constraint. <a href="#DP2026-abs-solved">[DP2026-abs-solved]</a>
Higher dimensions remain open. The case of general $d \geq 2$ is still open, although van Handel proved that some universal dimension-free comparator exists. <a href="#DP2026-dim-open">[DP2026-dim-open]</a> <a href="#vH25-thm11">[vH25-thm11]</a>
Partial higher-dimensional consequences. Davis and Power also prove a sequential tensorization principle for multivariate convex domination, and a dimension-free Gaussian comparator for the cone generated by convex ridge functions. <a href="#DP2026-abs-solved">[DP2026-abs-solved]</a>
A Strassen-type reformulation of van Handel's theorem is given as Corollary 1.2: one can construct $X$ and a standard Gaussian $G$ on a common space so that $X = c E [G ∣ X]$ . <a href="#vH25-cor12">[vH25-cor12]</a>
Historical discussion: MathOverflow question on sub-Gaussian variables and convex ordering.

References

<a id="DP2026"></a>[DP2026] Davis, Damek; Power, Sam. The sharp one-dimensional convex sub-Gaussian comparison constant. arXiv:2604.03170 (2026). DOI: 10.48550/arXiv.2604.03170. arXiv PDF: arXiv:2604.03170. Google Scholar
<a id="DP2026-abs-solved"></a>[DP2026-abs-solved] loc: arXiv PDF p.1, Abstract. quote: “Let $X$ be an integrable real random variable with mean zero and two-sided sub-Gaussian tail $P (∣ X ∣ > t) \leq 2 e^{- t^{2} /2}$ for all $t \geq 0$ . We determine the smallest constant $c_{⋆}$ such that $X$ is dominated in convex order by $c_{⋆} G$ , where $G$ is standard normal. Equivalently, $c_{⋆}^{2}$ is the sharp one-dimensional convex sub-Gaussian comparison constant appearing in the Optimization Constants in Mathematics repository [DITc26]. We show that $c_{⋆}$ is given by an explicit system of one-dimensional equations and is attained by an extremal distribution that saturates the tail constraint. Numerically, $c_{⋆} \approx 2.30952$ (so $c_{⋆}^{2} \approx 5.33386$ ).”
<a id="DP2026-def-tail"></a>[DP2026-def-tail] loc: arXiv PDF p.1, Section 1 “Setup and the constant”, equation (1). quote: “We call $X$ $1$ -sub-Gaussian in the tail sense if $E [X] = 0$ and $P (∣ X ∣ > t) \leq s_{G} (t) := min 1, 2 e^{- t^{2} /2}$ for all $t \geq 0$ .”
<a id="DP2026-def-constant"></a>[DP2026-def-constant] loc: arXiv PDF p.1, Section 1 “Setup and the constant”, equation (2). quote: “Define the one-dimensional comparison constant $c_{⋆} := in f c > 0 : every X satisfying (1) obeys X ⪯_{c x} c G$ , where $X ⪯_{c x} Y$ denotes convex domination: $E [f (X)] \leq E [f (Y)]$ for every convex $f : R \to R$ for which both expectations are finite.”
<a id="DP2026-thm1-sharp"></a>[DP2026-thm1-sharp] loc: arXiv PDF pp.1–2, Theorem 1. quote: “Theorem 1 (Sharp one-dimensional convex sub-Gaussian comparison). The sharp constant in (2) satisfies $c_{⋆} = c_{0}$ , where $c_{0}$ is defined by (3)–(6). In particular: 1. For every random variable $X$ satisfying (1) and every convex $f : R \to R$ , $E [f (X)] \leq E [f (c_{0} G)]$ whenever the right-hand side is finite. 2. For every $c < c_{0}$ , there exist a random variable $X^{⋆}$ satisfying (1) and a convex function $f$ such that $E [f (X^{⋆})] > E [f (c G)]$ . ... Consequently, the one-dimensional value of the constant $C_{48}$ in [DITc26] is $C_{48}^{(1)} = c_{0}^{2}$ .”
<a id="DP2026-rem2-num"></a>[DP2026-rem2-num] loc: arXiv PDF p.2, Remark 2. quote: “A direct high-precision evaluation of (3)–(6) gives $a \approx 1.80334$ , $p_{0} \approx 0.39342$ , $z \approx 0.27041$ , $c_{0} \approx 2.30952$ , $c_{0}^{2} \approx 5.33386$ . No numerical computation is used in the derivation of the exact characterization $c_{⋆} = c_{0}$ .”
<a id="DP2026-dim-open"></a>[DP2026-dim-open] loc: arXiv PDF p.2, end of Remark 3 / start of discussion after it. quote: “The case of general $d \geq 2$ remains open.”

<a id="vH25"></a>[vH25] van Handel, Ramon. On the subgaussian comparison theorem. arXiv:2512.18588 (2025). DOI: 10.48550/arXiv.2512.18588. arXiv PDF: arXiv:2512.18588. Google Scholar. Author PDF
<a id="vH25-thm11"></a>[vH25-thm11] loc: Author PDF p.1, Theorem 1.1. quote: “Let $X$ be any $1$ -subgaussian random vector in $R^{n}$ and $G \sim N (0, I_{n})$ be a standard Gaussian vector in $R^{n}$ . Then $E [f (X)] \leq E [f (c G)]$ for every convex function $f : R^{n} \to R$ , where $c$ is a universal constant.”
<a id="vH25-cor12"></a>[vH25-cor12] loc: Author PDF p.1, Corollary 1.2. quote: “There is a universal constant $c$ such that for every $1$ -subgaussian vector $X$ in $R^{n}$ , we can construct $X$ and a standard Gaussian vector $G \sim N (0, I_{n})$ on a common probability space such that $X = c E [G ∣ X]$ .”

What counts as progress

A better upper or lower bound, with a proof or a construction whose value is re-computed by published code (reproducible), ideally with a certificate a deterministic checker can validate.
A formal proof (Lean) of a known bound, or a precise error in a claimed one.
New references for the tables above (literature claims).

Source and licence

Imported from the crowdsourced repository of optimization constants (Terence Tao and contributors), commit 2c1968cd520b, Apache License 2.0; reformatted for this page. New records should also be reported there.