The Gaussian simplex measure conjecture
Let be the ball of radius centered at the origin, and let range over simplices containing this ball. Let the Gaussian measure have density proportional to . The Gaussian simplex measure conjecture. The Gaussian measure of is minimized at the regular simplex with inscribed ball . The claim is known in dimension by slicing and the result about spherical caps, but remains open in general.
References
Primary source
Alexey Balitskiy, Roman Karasev and Alexander Tsigler, “Optimality of codes with respect to error probability in Gaussian noise”, arXiv:1701.07986 (2017).
Progress summary
The three-dimensional case is proved, but the general conjecture remains open apart from an unverified 2026 submission claiming an equivalent solution.
The conjecture says that the regular simplex minimizes Gaussian measure among simplices containing a fixed ball. The general case remains unresolved.
Known results
- Dimension : proved by slicing and the spherical-cap result; the argument does not extend directly to higher dimensions (2017 source).
2026 developments
A 2026 paper claims a proof of the related Weak Simplex Conjecture, but states that its method does not establish the stronger Gaussian-measure conjecture. Separately, a September 9, 2026 community submission claims an equivalent stronger result that would imply this conjecture; the claim and accompanying Lean formalization are unverified.
Community submission (unverified)
On September 9, 2026, a submitted proof claimed a solution of an equivalent result, obtained with AI assistance, and pointed to a Lean formalization. No independent verification is recorded.
Current status (as of September 2026): The case is known; the general conjecture remains open, with only an unverified community claim of an equivalent proof.
Sources
- ar5iv.labs.arxiv.org
- arxiv.org
- arxiv.org
- arxiv.org
- math.stackexchange.com
- researchgate.net
- quantamagazine.org
- math.utah.edu
- numdam.org
- quantamagazine.org
- quantamagazine.org
- arxiv.org
- arxiv.org
- ar5iv.labs.arxiv.org
- mathstodon.xyz
- mathstodon.xyz
- mathstodon.xyz
- mathstodon.xyz
- quantamagazine.org
- scientificamerican.com
- quantamagazine.org
- arxiv.org
- eudml.org
- mathoverflow.net
- math.stackexchange.com
- wujns.edpsciences.org
- arxiv.org
- arxiv.org
- ar5iv.labs.arxiv.org
- export.arxiv.org
- mathstodon.xyz
- mathstodon.xyz
Solutions 1
ProofPreprint of a solution I have obtained with the help of AI. A Lean formalization is included as well.See full solution
I have posted a preprint of a solution to an equivalent result. I obtained it with the help of AI. A Lean formalization is included as well.