Convex and metric geometry·OpenAI Math·Result 101
The sharp simplex conjecture for isotropic constants
One of the 372 results in OpenAI's public mathematics catalogue, filed under convex and metric geometry: 1 preprint, the latest dated 5 October 2026. This page quotes OpenAI's own summary of the claim, links the paper and abstract, and embeds the animation.
In one line: OpenAI reports a proof of the sharp simplex conjecture for isotropic constants (convex and metric geometry) in a preprint dated 5 October 2026. It has not been peer reviewed. The channel's animation, Simplex Conjecture for Isotropic Constants: OpenAI Claims a Proof, is on YouTube.
What OpenAI claims
Proves that simplices uniquely maximize the isotropic constant among convex bodies in every dimension, resolving the strong isotropic constant conjecture. Also establishes the sharp entropy lower bound for log-concave probability densities, with equality precisely for invertible affine images of products of one-sided exponential laws.
Paper
A sharp entropy bound and the simplex inequality for isotropic constants
OpenAI's abstract We prove the strong isotropic constant conjecture: in each dimension, simplices are the unique maximizers of the isotropic constant among convex bodies. We also prove the sharp entropy bound h(f) ≥ m+1/2 log det Cov (f) for every log-concave probability density on ℝm, with equality precisely for invertible affine images of products of one-sided exponential laws.
Status
Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.
Quick answers
- Is it proven?
- Not independently yet. OpenAI posted a preprint on 5 October 2026; it has not been peer reviewed, so treat the result as OpenAI's claim until other mathematicians have checked it.
- What exactly does OpenAI claim?
- Proves that simplices uniquely maximize the isotropic constant among convex bodies in every dimension, resolving the strong isotropic constant conjecture. Also establishes the sharp entropy lower bound for log-concave probability densities, with equality precisely for invertible affine images of products of one-sided exponential laws.
- Where is the proof?
- In the paper, A sharp entropy bound and the simplex inequality for isotropic constants (PDF).
- Is there a video?
- Yes, on YouTube (0:20).
Also in
- Chapter 67 of 83 in Every OpenAI Math Result, Part 1 of 4: Goldfeld's Conjecture and 82 More · starts at 22:55 · watch from there on YouTube →
The video
Simplex Conjecture for Isotropic Constants: OpenAI Claims a Proof · Short · 0:20 · description, sources and transcript →
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