Submit progress

on One-dimensional convex sub-Gaussian comparison constant ($C_{48}$)

The statement

Let $X$ be an integrable real random variable. We say that $X$ is $1$ -sub-Gaussian in the tail sense if

\mathbf{E}[X]=0 \quad\text{and}\quad \mathbf{P}(\lvert X\rvert>t)\le 2e^{-t^2/2}\quad\text{for all }t\ge 0.

[DP2026-abs-solved] [DP2026-def-tail]

For real random variables $X,Y$ , write $X\preceq_{cx}Y$ if

\mathbf{E}[f(X)]\le \mathbf{E}[f(Y)]

for every convex $f:\mathbf{R}\to\mathbf{R}$ for which both expectations are finite. [DP2026-def-constant]

Let $G\sim\mathcal{N}(0,1)$ be standard normal. We define

C_{48}:=\inf\Bigl\{C>0:\ \forall\ \text{}1\text{-sub-Gaussian }X,\ X\preceq_{cx}\sqrt{C}\,G\Bigr\}.

[DP2026-def-constant]

Davis and Power proved that this one-dimensional problem is solved: if $c_\star$ denotes the sharp comparison factor, then $C_{48}=c_\star^2=c_0^2$ , where $c_0$ is determined by an explicit system of one-dimensional equations. Numerically,

C_{48}=c_0^2 \approx 5.33386.

[DP2026-abs-solved] [DP2026-thm1-sharp] [DP2026-rem2-num]

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.