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The Mahler Conjecture for General Convex Bodies

OpenAI · September 22, 2026

Lean

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Abstract

We resolve the Mahler conjecture for general convex bodies positively. For every convex body $K\subset\mathbb R^n$, n ≥ 1, with Santaló point $s(K)$, $|K|\,|(K-s(K))^\circ|\ge (n+1)^{n+1}/(n!)^2$, with equality exactly for simplices.

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work in progress

Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.

Comments and discussion

Formalization

  • OpenAI math release, Lean scope note for family 087 (lean/docs/087.md). States that the formalization proves $\inf_{z\in\mathrm{int}\,K}|K||(K-z)^\circ|\ge(n+1)^{n+1}/(n!)^2$ with equality exactly for simplices and excludes the functional inequality; another sentence of the same note says the nonsymmetric conjecture is not included. [family-level]
  • OpenAI math release, Theorem.lean (general_mahler) (openai/math, commit adc7f12). Lean statement of the sharp general inequality for compact convex bodies with nonempty interior, with equality exactly for simplices; the comparator challenge is GeneralMahler.lean.

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Cite

@misc{OAI:The-Mahler-Conjecture-for-General-Convex-Bodies-September-22-2026,
  author = {{OpenAI}},
  title = {{The Mahler Conjecture for General Convex Bodies}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-Mahler-Conjecture-for-General-Convex-Bodies-September-22-2026/paper.pdf}{OAI:The-Mahler-Conjecture-for-General-Convex-Bodies-September-22-2026}},
  year = {2026}
}

From OpenAI's README for this paper:

Verification

Use Python 3 without -O or PYTHONOPTIMIZE and the dependency in verification/requirements.txt (sympy==1.14.0). The runner executes the ten scalar and segment computation programs from unchanged source in a new temporary output directory:

(
  set -eu
  verification_tmp=$(mktemp -d)
  printf 'Verification results: %s\n' "$verification_tmp/run"
  python3 verification/reproduce.py --output "$verification_tmp/run"
)

Successful execution prints ten PASS lines and writes RUN.json with "passed": true. The output also contains per-program logs and generated reports.

Lean formalization

OpenAI provides Lean files for this family: lean/docs/087.md. The Lean statements have not been checked by people against the paper's claims.

Family 087: The Mahler conjectures, functional inequalities and polar-product symplectic width

Resolves the symmetric and nonsymmetric geometric Mahler conjectures in every dimension, with Hanner polytopes and simplices as the respective volume-product minimizers and all equality cases classified. The corresponding sharp functional Mahler inequalities also hold. For n ≥ 2, every symmetric polar product $K\times K^\circ$ in dimension $2n$ has Gromov width 4.

Other papers in this family: