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

1 · The essentials

These manuscripts study how small the combined size of a solid shape and its mathematically paired shape can be. The size being compared is the volume of one multiplied by the volume of the other. For shapes that look the same when reflected through the origin, the main manuscript gives an exact lower bound in every dimension and identifies every shape that reaches it. This result has a Lean formalization, meaning that it has been checked using a formal proof system. A companion manuscript states an exact lower bound without that symmetry requirement and identifies the shapes that reach it. Another Lean-formalized manuscript connects the symmetric bound to how balls can fit into a space made from the two paired shapes, under special rules for reshaping them. The repository also reports related inequalities for functions, but supplies no precise statements for those inequalities here.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine a shape and a companion shape linked by a fixed design rule, with the goal of making the product of their volumes as small as possible. The question is not just how small that product can get, but which designs attain the smallest value. This image leaves out both the actual pairing rule and the special rules governing the ball-fitting result.

Why it matters

The results give sharp bounds: they specify both the minimum volume product and all shapes attaining it, under the respective hypotheses. The ball-embedding manuscript also supplies a connection between a symplectic fitting problem and the symmetric volume inequality through volume preservation. The supplied material distinguishes formally verified results from companion claims whose formalization is not reported.

2 · Going further

The main problem concerns an origin-symmetric convex body $K \subset R^{n}$ and its polar body $K^{\circ}$ . Manuscript 1 establishes the symmetric Mahler bound $∣ K ∣ ∣ K^{\circ} ∣ \geq 4^{n} / n!$ , with equality exactly for invertible linear images of Hanner polytopes. It is marked Lean-formalized, including its stated equality classification. The source presents this as a positive resolution of the symmetric Mahler conjecture; it gives no account of earlier partial results.

Manuscript 3, also marked Lean-formalized, supplies the argument outline available in the abstracts. For every integer $n \geq 2$ , it constructs smooth symplectic embeddings of standard balls of every capacity $0 < c < 4$ into $int K \times int K^{\circ}$ , and states that this domain has Gromov width exactly $4$ . The abstract then invokes volume preservation to obtain the symmetric Mahler inequality in every dimension. It does not describe the embedding construction, the upper-bound argument for the width, or how the equality classification follows.

Manuscript 2 treats general convex bodies, using the Santaló point $s (K)$ to choose the center before taking the polar. It states that, for $n \geq 1$ , $∣ K ∣ ∣ (K - s (K))^{\circ} ∣ \geq (n + 1)^{n + 1} / (n!)^{2}$ , with equality exactly for simplices. This result is shown in the manuscript, but no Lean formalization is indicated for it. The repository summary additionally reports corresponding sharp functional Mahler inequalities, without supplying their formulas, hypotheses, equality cases, or arguments.

Useful notions

Convex bodies and volume
Symmetry about the origin
Polar bodies and volume products
Linear transformations
Symplectic embeddings and ball capacity

What was known before / what is new

Before

The source identifies the symmetric and general volume-product questions as Mahler conjectures. It does not provide historical dates, earlier bounds, or previously known special cases, so no more detailed account of prior knowledge can be drawn from it.

New

The main Lean-formalized manuscript establishes the symmetric lower bound and classifies all equality cases as invertible linear images of Hanner polytopes. The second manuscript states the general lower bound with simplices as exactly the equality cases. The third, also Lean-formalized, establishes Gromov width $4$ for symmetric polar products when $n \geq 2$ and connects ball embeddings to the symmetric inequality. Sharp functional counterparts are reported only in the repository summary.

Limits and assumptions

The main theorem requires an origin-symmetric convex body, while the general-body statement uses translation by the Santaló point. The symplectic statement concerns the product of the interiors of a symmetric body and its polar, and assumes $n \geq 2$ ; its construction provides embeddings for every capacity strictly below $4$ , not an asserted embedding at capacity $4$ . Lean formalization is indicated for Manuscripts 1 and 3, not for Manuscript 2 or the functional inequalities. The supplied abstracts do not give enough detail to reconstruct the arguments or assess the exact functional statements. They identify no remaining open question.

Mini glossary

Origin symmetry: A body is origin-symmetric when reflection through the origin leaves it unchanged.
Volume product: The quantity obtained by multiplying a body's volume by the volume of its polar, with the specified centering when needed.
Equality case: A shape for which the stated lower bound is attained exactly.
Hanner polytopes: The named family of polytopes whose invertible linear images are exactly the equality cases in the symmetric theorem.
Santaló point: The distinguished point $s (K)$ used to center a general convex body in the stated nonsymmetric volume-product inequality.
Gromov width: A measure of symplectic ball-fitting capacity, stated here to equal $4$ for the specified symmetric polar products.

Keywords : Mahler conjectures · Volume product · Gromov width

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

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 $2 n$ has Gromov width 4.

Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.

Scope of the Lean formalization (source text)

…

lean/docs/087.md

Manuscripts in this family

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The symmetric Mahler conjecture and its equality cases

22/09/2026

Verified by Lean

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

PDF GitHub folder Lean formalization

The Mahler Conjecture for General Convex Bodies

22/09/2026

To be verified

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

PDF GitHub folder

Symplectic Balls in Symmetric Polar Products

22/09/2026

Verified by Lean

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

PDF GitHub folder Lean formalization