How small can a convex body and its polar be together

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Family 087 · Convex and metric geometry
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-14

1. The question

A long thin convex body has small extent in some directions and large extent in others. Its polar reverses that relationship: directions in which the original body is narrow become directions in which the polar can extend far. Their volume product therefore measures shape in a way that ordinary volume does not.

Mahler’s problem asks how small this product can be. There are two versions. Among centrally symmetric bodies, cubes and their relatives are the proposed minimizers. Among arbitrary convex bodies, the proposed minimizers are simplices, the higher-dimensional analogues of triangles and tetrahedra.

This family contains three manuscripts. Two claim the sharp inequalities and complete equality classifications in every dimension. The third claims a stronger geometric embedding statement for the product of a symmetric body with its polar, viewed as a region in position–momentum space. Functional inequalities connect the same questions to convex functions and probability densities.

2. Background and essential definitions

A convex body $K\subset\mathbb R^n$ is compact, has nonempty interior, and contains the segment between any two of its points. Write $|K|$ for its $n$-dimensional volume. If zero lies in its interior, its polar is

$$K^\circ=\{y:\langle x,y\rangle\leq1\text{ for every }x\in K\}.$$

For an origin-symmetric body, $K=-K$, the product $|K||K^\circ|$ is unchanged by an invertible linear map: the two volumes acquire reciprocal determinant factors.

For a body without a prescribed center, polarity depends on where the origin is placed. The Santaló point $s(K)$ is the unique interior point minimizing the polar volume. The affine-invariant product is

$$P(K)=|K|\,|(K-s(K))^\circ|.$$

A simplex is the convex hull of $n+1$ affinely independent points. The symmetric candidate class is larger. Hanner polytopes start with centered intervals and are built recursively by taking Cartesian products or convex hulls of two bodies placed in complementary coordinate subspaces. Cubes and cross-polytopes are examples, but mixed constructions also occur.

For a concrete normalization, the cube $[-1,1]^n$ has volume $2^n$. Its polar is the cross-polytope $\{y:\sum_i|y_i|\leq1\}$, of volume $2^n/n!$. Their product is $4^n/n!$, the symmetric conjecture’s exact target.

3. Development history and previous partial results

Mahler’s work in the 1930s established the planar setting and connected volume products to the geometry of numbers. In the general planar problem, Meyer later completed the equality classification for arbitrary convex bodies: triangles are the minimizers.

Higher-dimensional sharp results were known under additional structure. Saint-Raymond proved the symmetric inequality for unconditional bodies, those unchanged by independently reversing coordinate signs. Meyer and Reisner identified the Hanner equality cases in that class. Zonoids, obtained as limits of sums of segments, also admitted the sharp symmetric inequality, with parallelotopes as their equality cases.

Iriyeh and Shibata proved the three-dimensional symmetric theorem. A 2026 preprint by Chen, Li, Xi and Xu treats the general three-dimensional problem as well. In other dimensions, shadow-system methods gave sharp results for polytopes with few vertices, and local theorems showed that simplices or Hanner bodies could not be improved by nearby perturbations. Local minimality is not a global classification.

A different line obtained the right exponential order without the exact constant. Bourgain and Milman’s reverse Santaló theorem was fundamental here. Kuperberg’s Gauss-linking approach, and complex-analytic work by Nazarov and others, developed additional methods. These estimates did not provide the sharp constants claimed in M1 and M2.

Symplectic geometry supplied another connection. Artstein-Avidan, Karasev and Ostrover computed a different capacity of symmetric polar products and related a proposed volume–capacity inequality to Mahler’s conjecture. Particular polar products were known to contain large symplectic balls. M3 claims the unrestricted symmetric-body version. It also explicitly notes that the unrestricted Viterbo volume–capacity conjecture was disproved by Haim-Kislev and Ostrover; the special polar-product theorem is not a revival of that general conjecture.

4. What the new manuscripts claim

The symmetric and general volume products

M1 states that every origin-symmetric convex body in every dimension $n\geq1$ satisfies

$$|K||K^\circ|\geq\frac{4^n}{n!},$$

with equality exactly for invertible linear images of Hanner polytopes.

M2 states that every convex body satisfies

$$P(K)\geq\frac{(n+1)^{n+1}}{(n!)^2},$$

with equality exactly for simplices. No smoothness or symmetry is assumed. Because the Santaló point minimizes the polar volume, the same lower bound holds when polarity is taken about any interior point.

These are separate sharp statements. The general lower constant is smaller than the symmetric one in dimensions at least two, so the general theorem alone does not prove the sharp symmetric theorem.

Functional consequences

For a proper lower-semicontinuous convex function $\varphi$, define its convex conjugate by

$$\varphi^*(y)=\sup_x\{\langle x,y\rangle-\varphi(x)\}.$$

“Proper” excludes an identically infinite function and values of negative infinity; positive infinity is allowed to restrict the support. The function $e^{-\varphi}$ is log-concave. Assuming its integral is finite and positive, the manuscripts claim

$$\left(\int e^{-\varphi}\right) \left(\int e^{-\varphi^*}\right) \geq \begin{cases} 4^n,&\varphi\text{ is even},\\ e^n,&\varphi\text{ is arbitrary}. \end{cases}$$

The second integral may be infinite. These consequences use earlier implications of Fradelizi and Meyer together with the geometric inequalities in all dimensions, not merely the dimension of the function.

Symplectic balls in polar products

M3 places $K$ in position coordinates $q$ and $K^\circ$ in momentum coordinates $p$, with symplectic form $\omega_0=\sum_jdq_j\wedge dp_j$. A symplectic embedding preserves this form, and hence volume, but is more constrained than an arbitrary volume-preserving map.

The standard ball of capacity $c$ is

$$B^{2n}(c)=\{(q,p):\pi(|q|^2+|p|^2)<c\}.$$

The Gromov width is the supremum of capacities of balls that admit such embeddings into a domain. M3 claims, for every symmetric $K$ and $n\geq2$,

$$c_G(\operatorname{int}K\times\operatorname{int}K^\circ)=4.$$

It constructs embeddings for every $c<4$; it does not require an embedding at the supremum. Since the ball’s volume is $c^n/n!$, volume preservation and passage to the limit yield the symmetric Mahler inequality. M3 does not classify volume-product equality cases; that is M1’s separate contribution.

5. Why the advance matters

Importance: 5/5. Sharp Mahler inequalities are central extremal problems in convex geometry. The claimed results would determine both the optimal constants and all minimizing shapes in arbitrary dimension, replacing broad asymptotic bounds and special-class theorems with exact global answers.

The equality classifications carry mathematical content beyond the numerical bounds. The general minimizer is a simplex, but the symmetric minimizers form a recursive family with different combinatorial shapes. Identifying all of them requires more than checking that a cube has the right product.

The symplectic statement adds a different kind of information. Having enough volume is ordinarily insufficient to fit a symplectic ball, because nonsqueezing imposes additional restrictions. A matching width theorem says that every symmetric polar product has a particular embedding capacity, regardless of corners, smoothness or strict convexity.

The functional consequences extend the significance from bodies to log-concave functions. Additional entropy–transport consequences in M1 and M2 concern distributions of gradients of convex potentials. Their hypotheses include essential continuity of the relevant log-concave densities; they are not unrestricted statements about every probability measure. The rating reflects the breadth of these claims, not an independent assessment of their proofs.

6. The main idea

Symmetry, a conformal lens, and missing volume

M1 first approximates a symmetric body by an intersection of finitely many symmetric strips. A special conformal map from a disk to a planar lens has a carefully chosen boundary distribution: a boundary coordinate becomes uniform.

This analytic construction assigns feasible simplices inside the polar body to points of the original body. A holomorphic zero-counting mass estimate gives a lower bound for an integral of their volumes. Since the simplices lie inside the polar, their total contribution cannot exceed the body–polar volume product. The normalization yields the sharp constant.

For equality, the integrated missing volume must disappear. The manuscript turns that vanishing deficit into a metric property of the associated normed space: triples admit points that simultaneously split pairwise distances additively. A classical classification of the corresponding ball-intersection structure then identifies the Hanner bodies.

General bodies, dual cones, and Gaussian projections

M2 lifts a convex body to a cone in one higher dimension. The volume-product inequality becomes a lower bound for a product of Laplace integrals over dual cones.

Nearest-point projections of biased Gaussian vectors supply paired maps into these cones. Their Jacobians, together with change of variables and entropy inequalities, reduce the desired estimate to a matrix problem. The projection derivatives at different layers need not commute, so estimating each layer independently would lose the information needed for the sharp bound.

The argument compares the whole family to a linear Gaussian matrix field, separates higher-order Gaussian components, and uses scalar inequalities on pairs of eigenvalues to control the remaining errors. In the claimed equality analysis, the matrices must commute and the cone splits into one-dimensional rays. A bounded section of that cone is a simplex.

Convert holomorphic vanishing into actual ball embeddings

M3 also uses the conformal lens, but pursues an embedding rather than a volume integral. High powers of a holomorphic coordinate create a zero of large order. A smoothing and deformation argument turns that order into a symplectic ball of a controlled capacity.

A separate uniform estimate places the constructed domain inside a product of the body and a scaled polar. The two scales have a ratio tending to four. Rescaling therefore fits every ball of capacity strictly below four into the original polar product. Approximation treats nonsmooth bodies, while a supporting-cylinder argument and nonsqueezing give the matching upper bound.

The three routes are related, but M3 does not assume the symmetric volume theorem or its equality classification. Its embedding claim is presented as an independent route to the numerical symmetric bound.

7. Scope, limitations, and open questions

All bodies are full-dimensional compact convex sets. The symmetric statements require symmetry about the origin used for polarity. The general statement uses the Santaló point; replacing it with a centroid without explanation would change the normalization of the problem.

The equality descriptions belong to the geometric inequalities. They should not be transferred automatically to convex potentials in the functional statements, whose extremizers have a different formulation.

The Gromov width is a supremum. Embeddings below capacity four suffice, and no boundary regularity is imposed on $K$. The theorem concerns a specific position–momentum product, not every convex region in symplectic space, and does not assert that all symplectic capacities agree there.

Quantitative stability asks how close a nearly minimizing body must be to an extremizer. Efficient ways to construct the symplectic embeddings are another distinct issue. Neither follows simply from the sharp constants and qualitative equality statements summarized here.

8. Sources and further reading

M1 covers the symmetric inequality, Hanner equality cases and even functional consequences. M2 treats the general inequality, simplex equality and general functional consequences. M3 supplies the polar-product width theorem and its volume deduction.

The linked low-dimensional papers and Kuperberg’s work give primary historical landmarks. Fradelizi and Meyer’s Some functional inverse Santaló inequalities supplies the earlier geometric-to-functional implications explicitly used by the manuscripts. Klartag’s Isotropic constants and Mahler volumes develops the cone–Laplace viewpoint used in M2.

Source manuscripts

• M1. OpenAI, The symmetric Mahler conjecture and its equality cases (September 22, 2026).
• M2. OpenAI, The Mahler Conjecture for General Convex Bodies (September 22, 2026).
• M3. OpenAI, Symplectic Balls in Symmetric Polar Products (September 22, 2026).

Sources consulted on 14 October 2026 at repository revision 1581d4c. Download the manuscript citations.