---
title: "The sharp simplex conjecture for isotropic constants"
authors:
  - "OpenAI (internal model)"
year: 2026
venue: "openai/math manuscript collection (GitHub), result family 101; unreviewed"
arxiv: null
doi: null
source: "https://github.com/openai/math/blob/main/preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/isotropic-simplex.pdf"
topics:
  - "openai-math"
  - "ai-generated-math"
  - "convex-and-metric-geometry"
  - "unformalized"
seed_rank: 1958
seed_batch: "openai-math-2026-10-07"
reviewed: "2026-10-07"
pool: "maths-foundations"
relevance_score: 7
lineage: ai-mathematical-reasoning
cites:
  - title: "A sharp entropy bound and the simplex inequality for isotropic constants"
    url: "https://github.com/openai/math/blob/main/preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/isotropic-simplex.pdf"
    year: 2026
    arxiv: null
    doi: null
---

# The sharp simplex conjecture for isotropic constants

## One-sentence takeaway

OpenAI's result family 101 (Convex and metric geometry) claims: Proves that simplices uniquely maximize the isotropic constant among convex bodies in every dimension, resolving the strong isotropic constant conjecture.

## Why it matters here

Convex and metric geometry underlies collision, packing, embedding and spatial data-structure work. More broadly, every card in this batch is evidence for the `ai-mathematical-reasoning` thread: per the repo README, an unreleased internal OpenAI model produced it, using about three hours of ChatGPT Pro thinking compute per result on average, across roughly 4,000 posed problems.

## Key ideas

- Family summary (repo): 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.
- *A sharp entropy bound and the simplex inequality for isotropic constants*: 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 $h(f)\ge m+\tfrac12\log\det\mathop{\mathrm{Cov}}\nolimits (f)$ for every log-concave probability density on ℝm, with equality precisely for invertible affine images of products of one-sided exponential laws.

## Caveats

- Unreviewed: these are AI-generated manuscripts posted to GitHub with no peer review or arXiv deposit. The repo README says results are "at different stages of verification" and that "some of the unformalized results could have issues."
- This card restates the repo's family summary, manuscript abstracts and Lean scope notes. It is not an independent check of the proofs, and the scope of each theorem is exactly as stated there.
- Lean: not formalized. The repo has no Lean development for this family, so the claim rests on the prose manuscript(s) alone.

## Links

- Manuscript: [A sharp entropy bound and the simplex inequality for isotropic constants](https://github.com/openai/math/blob/main/preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/isotropic-simplex.pdf)
- Family entry: https://github.com/openai/math/blob/main/CONTENTS.md
- Overview PDF: https://github.com/openai/math/blob/main/overview.pdf
- Repo: https://github.com/openai/math