Result 087, Convex and metric geometry

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×K∘K\times K^\circ in dimension 2n2n has Gromov width 4.

Lean formalization Reasoning summary (PDF) Proof

The bigger picture

Why it matters

How small can a convex shape and its geometric dual be at the same time? These unreviewed manuscripts claim exact answers in every dimension, identifying the shapes that achieve the smallest combined volumes.

What changes?

For every dimension n at least one, the manuscripts claim sharp lower bounds on a convex body's volume times its polar's volume. A convex body is a compact convex set with interior; its polar consists of vectors having dot product at most one with all its points. Origin-symmetric bodies have bound 4^n/n!, attained exactly by invertible linear images of Hanner polytopes. General bodies, centered at their volume-product-minimizing Santaló point, have bound (n+1)^(n+1)/(n!)^2, attained exactly by simplices.

What does that help mathematicians do?

The symplectic claim connects ball fitting to volume bounds. For every origin-symmetric K in dimension n at least two, the interior of K times its polar, in dimension 2n, reportedly has Gromov width four. This width is the supremum of ball capacities embeddable by transformations preserving symplectic structure. The manuscript constructs embeddings for every capacity strictly between zero and four; their volume preservation is reported to yield the symmetric Mahler bound in every dimension.

Are there practical applications?

The immediate value is foundational: the equality classifications identify exactly which shapes can attain the smallest volume products, ruling out other proposed minimizers. The summary also reports corresponding sharp functional Mahler inequalities, linking this extremal picture to inequalities for functions. These are claimed mathematical consequences, not demonstrations of a practical technology.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

3 manuscripts

The symmetric Mahler conjecture and its equality cases

September 22, 2026 26 pages Main result formalized in Lean

We resolve the symmetric Mahler conjecture positively, including its equality classification. Every origin-symmetric convex body in ℝn has volume product at least 4n/n!4^n/n!, with equality exactly for invertible linear images of Hanner polytopes.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:The-symmetric-Mahler-conjecture-and-its-equality-cases-September-22-2026,
  author = {{OpenAI}},
  title = {{The symmetric Mahler conjecture and its equality cases}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-symmetric-Mahler-conjecture-and-its-equality-cases-September-22-2026/paper.pdf}{OAI:The-symmetric-Mahler-conjecture-and-its-equality-cases-September-22-2026}},
  year = {2026}
}

The Mahler Conjecture for General Convex Bodies

September 22, 2026 72 pages

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

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Symplectic Balls in Symmetric Polar Products

September 22, 2026 19 pages Main result formalized in Lean

For every integer n ≥ 2 and every origin-symmetric convex body K⊂RnK\subset\mathbb R^n, we prove that the Gromov width of intK×intK∘\mathop{\mathrm{int}}\nolimits K\times\mathop{\mathrm{int}}\nolimits K^\circ is 4. We construct smooth symplectic embeddings of standard balls of every capacity 0<c<40\lt c\lt 4 into this polar product. Volume preservation then resolves the symmetric Mahler conjecture positively in every dimension.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Symplectic-Balls-in-Symmetric-Polar-Products-September-22-2026,
  author = {{OpenAI}},
  title = {{Symplectic Balls in Symmetric Polar Products}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Symplectic-Balls-in-Symmetric-Polar-Products-September-22-2026/paper.pdf}{OAI:Symplectic-Balls-in-Symmetric-Polar-Products-September-22-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/087.md.

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

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The symmetric Mahler conjecture predicts ∣K∣∣K∘∣≥4n/n!|K||K^\circ|\ge4^n/n! for every origin-symmetric convex body K⊂RnK\subset\mathbb R^n. The formalization establishes this for every n≥1n\ge1 and characterizes equality exactly by invertible linear images of Hanner bodies, built from intervals using Cartesian products and convex-hull joins.

The nonsymmetric Mahler conjecture and the functional inequalities are not included.

The general Mahler conjecture gives a sharp lower bound for the volume product of a convex body and its polar. For every n≥1n\ge1 and compact convex body K⊂RnK\subset\mathbb R^n with nonempty interior, the formalization proves inf⁡z∈int K∣K∣ ∣(K−z)∘∣≥(n+1)n+1/(n!)2\inf_{z\in\mathrm{int}\,K}|K|\,|(K-z)^\circ|\ge (n+1)^{n+1}/(n!)^2. Equality holds exactly when KK is a simplex. The paper's functional inequality is outside this selected statement.

For an origin-symmetric convex body K⊂RnK\subset\mathbb R^n, n≥2n\ge2, the formalized result determines the symplectic ball capacity of int K×int K∘\mathrm{int}\,K\times\mathrm{int}\,K^\circ: its Gromov width is 44, and every ball of capacity 0<c<40<c<4 embeds symplectically into it. The normalization assigns capacity πr2\pi r^2 to a ball of radius rr.

No boundary smoothness or strict convexity is assumed. An embedding at capacity exactly 44 is not asserted.

Comparator links

Result Comparator statement
Symmetric Mahler inequality MahlerConjecture.lean
Symmetric Mahler equality characterization SymmetricMahlerEquality.lean
General Mahler inequality and simplex equality cases GeneralMahler.lean
Gromov width of symmetric polar products SymmetricPolar.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.