Mahler volume products: symmetric bodies, simplices, and symplectic balls

Source and verification

This article explains family 087 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

🏷️ Convex duality and the volume product

A convex body is a compact convex set with nonempty interior. If $0$ lies in its interior, its polar is

$$K^{\circ }=\{y:\underset{x\in K}{\sup }⟨x,y⟩\le 1\}.$$

The polar exchanges long directions with short directions. For an invertible linear map $T$, one has $(TK{)}^{\circ }=T^{-T}{K}^{\circ }$. Consequently the Jacobian factors cancel:

$$|TK||(TK{)}^{\circ }|=|K||K^{\circ }|.$$

Here $|K|$ denotes $n$-dimensional volume. Thus the volume product measures shape rather than size or the choice of linear coordinates. This family has three approaches: a symmetric volume-product inequality, the corresponding nonsymmetric problem after translation, and a symplectic embedding theorem. (OpenAI, 2026) (OpenAI, 2026a) (OpenAI, 2026a)

🏷️ Two different sharp inequalities

Symmetric statement in the first manuscript

For every origin-symmetric convex body $K\subset {\mathbb{R}}^n$,

$$|K||K^{\circ }|\ge \frac{4^n}{n!}.$$

Equality is claimed precisely for invertible linear images of Hanner polytopes. (OpenAI, 2026c)

Hanner polytopes start with a symmetric interval. One recursively takes either Cartesian products or the convex hull of two bodies placed in complementary coordinate subspaces. These are the unit-ball operations associated with the maximum norm and the sum norm.

The cube supplies an elementary check. Its polar is the cross-polytope:

$$([-1,1{]}^n{)}^{\circ }=\left\{y:\sum _{j=1}^n|y_j|\le 1\right\}.$$

The cube has volume $2^n$. In each orthant the cross-polytope is a simplex of volume $1/n!$, so its total volume is $2^n/n!$. Their product is exactly the proposed constant.

For a nonsymmetric body the origin must be chosen. The Santaló point $s(K)$ is the interior point minimizing $|(K-z{)}^{\circ }|$ over translations $z$. The second manuscript claims

$$|K||(K-s(K){)}^{\circ }|\ge \frac{(n+1{)}^{n+1}}{(n!{)}^2},$$

with equality exactly for simplices. (OpenAI, 2026b) The two constants differ because the two optimization classes differ. Applying the general inequality to a symmetric body does not recover the stronger symmetric constant.

🏷️ The symmetric proof mechanism

The first manuscript initially works with a symmetric polytope written as an intersection of slabs. It uses holomorphic maps into planar lenses to encode which slab constraints are active. An active collection determines a simplex ${\Sigma }_X$ contained in the polar body. The crucial analytic assertion is a lower bound on a total mass $S$, obtained from isolated zeros and a Stokes argument:

$$\begin{array}{rl}1\le S& =\frac{n!}{4^n}{\int }_K|{\Sigma }_X|dX\\ & \le \frac{n!}{4^n}|K||K^{\circ }|.\end{array}$$

The final inequality only uses ${\Sigma }_X\subseteq K^{\circ }$. All the sharpness is therefore concentrated in the first inequality and its exact normalization. This explains why a rough estimate for the lenses would not suffice: any fixed loss would propagate directly to the volume-product constant. (OpenAI, 2026c)

Equality needs more than the volume estimate. The manuscript carries a zero-deficit condition through an additional interval construction and obtains a metric median property of the polar unit ball. It then uses the classification associated with that property to identify Hanner polytopes. Approximation extends the inequality beyond polytopes; the equality argument must separately retain the rigidity information.

The source also discusses an even functional version involving a convex function and its Legendre transform. Its transfer uses a separate all-dimensional principle; it is not an additional direct consequence of the displayed integral alone.

🏷️ The nonsymmetric cone argument

The second manuscript passes from a translated body to the cone

$$C=\{(tx,t):t\ge 0,x\in K-z_0\}.$$

For a cone, the natural analogue of volume is a Laplace integral ${\chi }_C(v)={\int }_C{e}^{-⟨v,x⟩}dx$, defined for suitable dual vectors. The claimed sharp cone inequality couples this integral with that of the positive dual cone. The normalization fixes the pairing of the two vectors at $n+1$. Slicing the cone then returns the nonsymmetric volume product. (OpenAI, 2026b)

The analytic construction uses projections onto dual cones and Gaussian variables with a covariance chosen by a critical-point condition. The covariance is a matrix, and its interaction with the varying field cannot be reduced at the outset to scalar inequalities in a common eigenbasis. The manuscript retains these noncommuting terms, controls the higher Gaussian-chaos components, and isolates scalar inequalities with strict remainders.

In the proposed equality analysis, vanishing of those remainders forces a decomposition into one-dimensional cone factors. A height-one section of such a cone is a simplex. Some scalar bounds are supported by rational and interval certificates in the source; those certificates were not executed for this article.

🏷️ Symplectic balls and the same constant

On ${\mathbb{R}}_q^n\times {\mathbb{R}}_p^n$, the standard symplectic form is $\omega ={\sum }_{j}d{q}_j\wedge d{p}_j$. A symplectic map preserves this form and hence preserves $2n$-dimensional volume. A ball of radius $r$, written in terms of its capacity $c=\pi r^2$, has volume $c^n/n!$.

The third manuscript claims that the Gromov width of the interior of $K\times K^{\circ }$ is $4$ for origin-symmetric $K$ and $n\ge 2$. In particular, for every $c<4$ it constructs a symplectic embedding of a ball of capacity $c$. (OpenAI, 2026a) Therefore

$$|K||K^{\circ }|\ge \frac{c^n}{n!}(c<4).$$

Letting $c\uparrow 4$ gives the symmetric volume inequality. An embedding at the endpoint $c=4$ is not required or asserted by this limiting deduction. Dimension one can be checked directly using intervals.

The construction relates holomorphic vanishing order to embedded ball capacity. Lens estimates must remain uniform near their tips, and a Moser deformation is performed on compact sublevels without assuming every level is regular. A containing symplectic cylinder and nonsqueezing provide the upper bound for the width. This route supplies the sharp inequality, while the Hanner equality classification belongs to the first manuscript.

🏷️ Key steps and proof outlines

Holomorphic multiplicity supplies the sharp symmetric constant

For a symmetric polytope given by slab inequalities, the lens map converts each slab coordinate into a holomorphic function bounded by one. Raise these functions to a large common power $k$ and combine them into a vector-valued holomorphic map. Its isolated zero has a prescribed vanishing order. The complex Hessians of both the squared modulus and its logarithm are positive semidefinite away from that zero.

Stokes’ theorem compares the mass near the zero with mass near the outer level. Cauchy–Binet expands the top exterior power of the Hessian into squared Jacobian minors. Each minor corresponds to a choice of active slab coordinates. In the limit $k\to \infty$, the active coordinates lie on the lens boundary. The carefully normalized boundary law then converts the complex integral into a real simplex-volume integral. This is the chain producing $1\le S$, with exactly the constant in the earlier display. (OpenAI, 2026c)

The terminal comparison $|{\Sigma }_X|\le |K^{\circ }|$ is elementary. The source’s difficult contribution is the normalized mass-to-simplex identity before it. Losing a constant at the lens boundary or in the Jacobian conversion would alter the claimed sharp inequality.

Equality requires a pointwise witness beyond integral saturation

Equality in the volume estimate first says that a nonnegative missing-volume deficit has zero integral. That conclusion is initially almost everywhere and depends on the chosen polytope approximation. The zero-deficit section constructs a common feasibility witness and propagates the vanished deficit to each relevant point and direction. Only then can the interval construction derive a metric median property.

For three points in the polar norm, a median is a point lying simultaneously on geodesic intervals between each pair, where an interval means equality in the triangle inequality. The source converts the feasibility witness into such a common point. Classification of the resulting norm geometry identifies Hanner bodies. This separate argument explains why approximation of the inequality alone would not classify equality for arbitrary nonsmooth bodies.

Cone slicing reduces the nonsymmetric problem exactly

Let $m=n+1$, $C=\{(tx,t):t\ge 0,x\in K-z_0\}$, and let $D$ be its positive dual. The height-$t$ slice of $C$ has volume $t^n|K|$, while that of $D$ is a reflected copy of $t(K-z_0{)}^{\circ }$. Therefore

$${\chi }_C((0,1))=n!|K|,{\chi }_D((0,m))=n!m^{-m}|(K-z_0{)}^{\circ }|.$$

The factorial follows from ${\int }_0^{\infty }{t}^n{e}^{-t}dt=n!$; the power $m^{-m}$ follows by rescaling $t$. The cone pairing is $m$, exactly the normalization needed by the sharp cone inequality. This calculation shows where the nonsymmetric constant comes from rather than treating it as a variant of the symmetric one. (OpenAI, 2026b)

The projection construction uses Moreau’s identity with the positive-dual sign convention,

$${\Pi }_C(w)-{\Pi }_D(-w)=w,⟨{\Pi }_C(w),{\Pi }_D(-w)⟩=0.$$

Projection variational inequalities prove this identity and the one-Lipschitz property. The manuscript then introduces biased Gaussian inputs and chooses a simultaneous matrix normalization by a variational condition. Entropy change of variables produces log-Jacobian terms; Gaussian integration by parts and chaos decompositions separate the linear contribution from higher-order errors. Scalar inequalities with strict remainders bound those errors. Vanishing of the remainders forces one-dimensional cone factors, whose section is a simplex. The matrix normalization cannot be replaced by separate scalar normalizations unless simultaneous diagonalization has actually been justified.

The symplectic route turns vanishing order into room for a ball

The third manuscript uses holomorphic vanishing to construct a symplectic ball inside a suitable sublevel. A Moser deformation transports the symplectic form through a family while controlling the flow on compact sublevels. Lens slice estimates then realize that sublevel inside $K\times K^{\circ }$. Uniformity near lens tips is needed because a compact exhaustion approaches those singular boundary regions.

For every capacity $c<4$, the construction supplies an embedding; volume preservation gives the symmetric inequality by letting $c\uparrow 4$. Approximation passes from smooth or polyhedral models to arbitrary symmetric bodies. A containing cylinder and nonsqueezing give the reverse capacity bound, hence width exactly four. This capacity equality does not itself recover the Hanner classification, which belongs to the distinct zero-deficit argument. (OpenAI, 2026a)

See Also

Projection-body extrema studies another affine-invariant volume functional, whose minimum and maximum require different arguments. Entropy and isotropic constants uses a related cone construction to turn an analytic inequality into a simplex extremum.

🏷️ Manuscripts and verification scope

• The symmetric Mahler conjecture and its equality cases — pinned manuscript and companion source.
• The Mahler Conjecture for General Convex Bodies — pinned manuscript and companion source.
• Symplectic Balls in Symmetric Polar Products — pinned manuscript and companion source.

The family scope page describes selected formalization coverage. The scope describes the symmetric inequality and Hanner equality, the general inequality and simplex equality, and the symplectic embedding theorem. It does not include the functional extension. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026a. Symplectic Balls in Symmetric Polar Products.

🐻  OpenAI 2026b. The Mahler Conjecture for General Convex Bodies.

🐻  OpenAI 2026c. The symmetric Mahler conjecture and its equality cases.