| # The Mahler conjectures, functional inequalities and polar-product symplectic width |
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| The following describes the scope of the Lean formalization related to the following accompanying paper(s): |
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| - [The symmetric Mahler conjecture and its equality cases](../../preprints/The-symmetric-Mahler-conjecture-and-its-equality-cases-September-22-2026/paper.pdf) |
| - [The Mahler Conjecture for General Convex Bodies](../../preprints/The-Mahler-Conjecture-for-General-Convex-Bodies-September-22-2026/paper.pdf) |
| - [Symplectic Balls in Symmetric Polar Products](../../preprints/Symplectic-Balls-in-Symmetric-Polar-Products-September-22-2026/paper.pdf) |
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| ## Scope |
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| The symmetric Mahler conjecture predicts $|K||K^\circ|\ge4^n/n!$ for every origin-symmetric convex body $K\subset\mathbb R^n$. The formalization establishes this for every $n\ge1$ and characterizes equality exactly by invertible linear images of Hanner bodies, built from intervals using Cartesian products and convex-hull joins. |
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| The nonsymmetric Mahler conjecture and the functional inequalities are not included. |
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| The general Mahler conjecture gives a sharp lower bound for the volume product of a convex body and its polar. For every $n\ge1$ and compact convex body $K\subset\mathbb R^n$ with nonempty interior, the formalization proves |
| $\inf_{z\in\mathrm{int}\,K}|K|\,|(K-z)^\circ|\ge (n+1)^{n+1}/(n!)^2$. |
| Equality holds exactly when $K$ is a simplex. The paper's functional inequality is outside this selected statement. |
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| For an origin-symmetric convex body $K\subset\mathbb R^n$, $n\ge2$, the formalized result determines the symplectic ball capacity of $\mathrm{int}\,K\times\mathrm{int}\,K^\circ$: its Gromov width is $4$, and every ball of capacity $0<c<4$ embeds symplectically into it. The normalization assigns capacity $\pi r^2$ to a ball of radius $r$. |
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| No boundary smoothness or strict convexity is assumed. An embedding at capacity exactly $4$ is not asserted. |
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| ## Comparator links |
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| | Result | Comparator statement | |
| | --- | --- | |
| | Symmetric Mahler inequality | [MahlerConjecture.lean](../ComparatorChallenges/MahlerConjecture.lean) | |
| | Symmetric Mahler equality characterization | [SymmetricMahlerEquality.lean](../ComparatorChallenges/SymmetricMahlerEquality.lean) | |
| | General Mahler inequality and simplex equality cases | [GeneralMahler.lean](../ComparatorChallenges/GeneralMahler.lean) | |
| | Gromov width of symmetric polar products | [SymmetricPolar.lean](../ComparatorChallenges/SymmetricPolar.lean) | |
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