For every integer n≥ 2, let X=(X_1,…,X_n) be a centered Gaussian vector satisfying operatornameVar(X_i)=1 for every i. Let Z_1,…,Z_n be independent standard Gaussian random variables and set overlineZ=(Z_1+…+Z_n)/n. Then, for every t∈R, Pmax_i X_i≤ t≥ P√fracnn-1,max_i(Z_i-overlineZ)≤ t. Moreover, for each fixed t>0, equality holds if and only if operatornameCov(X_i,X_j)=-1/(n-1) for all i≠ j.

Full Simplex Conjecture

For every integer $n\ge 2$ , let $X=(X_1,\ldots,X_n)$ be a centered Gaussian vector satisfying $\operatorname{Var}(X_i)=1$ for every $i$ . Let $Z_1,\ldots,Z_n$ be independent standard Gaussian random variables and set $\overline{Z}=(Z_1+\cdots+Z_n)/n$ . Then, for every $t\in\mathbb{R}$ , $\mathbb{P}\{\max_i X_i\le t\}\ge \mathbb{P}\left\{\sqrt{\frac{n}{n-1}}\,\max_i(Z_i-\overline{Z})\le t\right\}$ . Moreover, for each fixed $t>0$ , equality holds if and only if $\operatorname{Cov}(X_i,X_j)=-1/(n-1)$ for all $i\ne j$ .

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Geometric Gaussian-measure formulation
Among all simplices containing a given centered Euclidean ball, the regular simplex circumscribed about the ball has the least standard Gaussian measure.
source: Stochastic Domination of Gaussian Maxima by the Regular Simplex

References

Primary source

arXiv

Additional references

Stochastic Domination of Gaussian Maxima by the Regular Simplex — arXiv — Abhijeet Mulgund

Posted 1 week ago · edited 1 week ago

Progress summary

Refreshed 1 week ago

Claimed solved

A September 2026 preprint claims to prove the conjecture, including uniqueness, but the result is unrefereed and unverified.

The conjecture asserts that the regular simplex gives the sharp Gaussian-maxima bound and the optimal simplex signaling performance. No original proposer or date was identified in the retrieved sources.

Known results

A 2020 paper claimed the four-dimensional Gaussian maximum result and the three-dimensional Simplex Mean Width Conjecture.
A 2021 paper gave an elementary proof of the three-dimensional mean-width case and recorded the four-dimensional Gaussian result.

September 2026 claimed completion

Abhijeet Mulgund's September 2026 preprint claims the all-dimensional unsmoothed comparison, characterizes equality, and derives uniqueness of the regular simplex for Gaussian signal identification and related coding problems. Earlier 2026 versions established the bound but left equality open; the newer claim says that gap is closed. No independent verification or referee report was found.

Current status (as of September 2026): the conjecture is claimed solved, including its equality cases, but the proof remains unverified; no unresolved mathematical case is established by the retrieved evidence.

Sources

No solutions have been posted yet.

Geometric Gaussian-measure formulation

Primary source

Additional references

Progress summary