OpenAI's Mahler claim: the route through an 87-year-old problem

On this page

Finding a promising shape and proving that no shape can do better are different achievements. The second can change what an entire field is able to build on. OpenAI's Mahler claim asks us to consider AI systems contributing at that level, supplying an exact rule across every admissible shape and every dimension. We think its lasting significance would be the mathematics other researchers could develop from that result and its proof.

The human history makes the size of that claim clear. In 1939, Kurt Mahler asked how small the volume product of a centrally symmetric convex shape and its polar could be. Eighty-seven years later, mathematicians had proved important special cases and universal estimates. The exact lower bounds in every dimension, together with every shape attaining them, remained the challenge.

OpenAI's 6 October mathematics release attributes proposed solutions to an internal model. We followed the earlier geometric methods, the new analytic constructions and the released reasoning summary to understand how the approach changed. A square and a triangle let us see the problem. The attempted universal proofs let us examine what a machine contribution could add to a research community that has worked on it for generations.

The problem Mahler left behind

Imagine a filled shape with no dents: any straight line between two points in it stays inside it. That is the convexity condition. Choose an interior point as the origin, then construct a second shape, its polar, using inequalities that encode the original's reach in every direction. Multiply their volumes. Which shapes make that product smallest?

Kurt Mahler's lower-bound conjecture dates to 1939. It asks for an answer across dimensions, where drawing a few promising shapes can only take us so far. The historical survey by Fradelizi, Meyer and Zvavitch also explains an intriguing imbalance: after choosing the appropriate centre, work by Wilhelm Blaschke and Luis Santaló established the upper bound, reached by ellipsoids. The minimum proved much more resistant.

Jean Bourgain and Vitali Milman's 1987 work established a universal lower estimate, without the sharp constant Mahler's problem demands. Another route was to solve the geometry exactly in a particular dimension. Hiroshi Iriyeh and Masataka Shibata settled the symmetric three-dimensional case, including equality. Their preprint appeared in 2017; the journal paper followed in 2020.

Their method divides a body and its polar into eight corresponding pieces. It seeks a linear change of coordinates that balances the relevant volumes and face areas, with a topological argument guaranteeing that such a balance exists. Once the pieces line up in the required way, the estimates give the sharp three-dimensional bound. The achievement is exact, but the construction is three-dimensional.

A 2026 preprint by Chen, Li, Xi and Xu addresses the general three-dimensional problem through a different deformation method. It moves vertices of a polytope while keeping existing faces planar, and uses the available motions to force a minimiser towards a tetrahedron. The new OpenAI manuscripts ask for an argument across dimensions, where those particular three-dimensional constructions do not supply the result.

This is the obstacle behind “every dimension”. A non-sharp estimate reaches all bodies but misses the target constant. A sharp three-dimensional argument reaches the constant but leaves higher dimensions open. OpenAI claims to obtain the sharp bounds and their equality cases together, without restricting the dimension.

Stretch the square and the product doesn't change

Take the square $[-1,1]^2$. Its area is 4. Its polar is the diamond with vertices (1,0), (0,1), (−1,0) and (0,−1), with area 2. The volume product is 8.

The polar comes from a simple rule: a vector $y$ belongs to it when $x\cdot y\le 1$ for every $x$ in the original body. For our square, that becomes $|y_1|+|y_2|\le 1$. Each point of the original imposes a restriction on the polar. Fradelizi, Meyer and Zvavitch's survey supplies the definitions and classical background.

Now stretch the square horizontally by a positive factor $a$. Its area becomes $4a$. The polar compresses horizontally by $1/a$, so its area becomes $2/a$. The product stays 8.

More generally, an invertible linear transformation multiplies one volume by its absolute determinant and the polar volume by the reciprocal. We found this a useful entrance to the problem. Size and linear coordinates disappear from the comparison. Even an extremely elongated centred parallelogram has the same product as the square.

What remains is the shape. A convex body means a compact convex subset of $\mathbb{R}^n$ with nonempty interior; these are ordinary Euclidean volumes. The question is how small the product can become across all the permitted shapes.

The cube has company

For origin-symmetric bodies, $K=-K$, OpenAI's symmetric manuscript claims:

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

Here $K^\circ$ is the polar, $|K|$ is the volume of the original body, and $n$ is the dimension.

Our square reaches the two-dimensional value. In n dimensions, a cube has volume $2^n$ and its polar cross-polytope has volume $2^n/n!$, giving the same formula.

But the cube is one member of a larger family. Hanner polytopes are built from centred intervals using Cartesian products or convex hulls of bodies placed in complementary coordinate spaces. These operations produce cubes, cross-polytopes and mixtures. They all share the target product.

Jaegil Kim established strict local minimality at Hanner polytopes among symmetric bodies. That controls nearby shapes. OpenAI claims a global bound and equality precisely for invertible linear images of the Hanner family.

We think the classification belongs in the foreground. Showing a cube reaches the value leaves two enormous questions: can some other shape do better, and can an unexpected shape tie? The claimed theorem must answer both.

A triangle changes the answer

Remove symmetry and even the choice of centre becomes part of the problem. Moving the origin changes the polar. The Santaló point is the unique interior point $z$ minimising $|K|\,|(K-z)^\circ|$. It generally isn't the original body's centre of mass; its defining condition centres the polar.

We checked a triangle with vertices (1,0), (0,1) and (−1,−1). Its area is 3/2. Its polar at the origin has vertices (1,1), (−2,1) and (1,−2), with area 9/2. The polar's centroid is the origin, confirming the Santaló choice. The product is 27/4, or 6.75.

That is below the symmetric value of 8. OpenAI's general manuscript claims:

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

The term $s(K)$ denotes the Santaló point: we move that point to the origin before taking the polar.

Equality would hold exactly for simplices, the convex hulls of n+1 affinely independent points. In the plane, those are triangles. Kim and Reisner had already proved strict local minimality at simplices. Again, the claimed advance is universal.

This is why we read the papers as two arguments. The general lower bound is weaker than the symmetric one, even in two dimensions. Proving it wouldn't automatically settle the sharper symmetric question.

The exact constant makes the proof unforgiving

Both manuscripts try to make geometry accessible to analytic estimates, then use the conditions for equality to recover the winning shapes. The sharp constant is essential: an estimate that loses a factor at an intermediate step can miss the conjecture even if it is otherwise powerful.

The symmetric paper's overview changes the comparison. It uses an existing conformal map from a disk to a lens-shaped region to construct probability laws. Those laws assign contributions to simplices—higher-dimensional triangles—that fit inside the polar. The paper adds the contributions into a quantity that its analytic argument forces to be at least 1.

Geometry supplies the other side: an exact integral comparison bounds that quantity above by the volume product multiplied by $n!/4^n$. If both steps hold, the volume product must be at least $4^n/n!$. The intended gain over the earlier three-dimensional balancing argument is a sharp comparison that can be made in arbitrary dimension. The complex-analytic tool is inherited; the claimed advance lies in the construction and exact comparison built from it.

Equality requires additional reasoning. If the volume product reaches the minimum, every inequality used along the way has to permit equality. The manuscript extracts a special metric structure from that requirement and connects it to the Hanner family. This is why a sharp bound and a classification of all minimisers are separate achievements.

The general paper instead lifts the body into a cone one dimension higher. A bounded slice recovers a convex body, allowing the volume product to be written through paired integrals on the cone and its dual.

That change of setting was already available. Bo'az Klartag's work uses Laplace transforms of dual cones and connects the sharp Mahler bound to another unresolved sharp inequality. OpenAI's proposed addition is the comparison of those integrals. It does not invent the cone reformulation.

The argument pairs nearest-point projections onto the dual cones and averages using Gaussian distributions, the higher-dimensional version of a bell-shaped distribution. The construction uses a varying bias to fix the projected mean. It chooses the Gaussian input's covariance—the spread and relationships among directions—together with the coordinates so the later matrix cancellation works. Those choices are meant to retain the sharp inequality where a less carefully matched average would lose information.

The derivative matrices need not commute: changing the order of two matrix operations can change the result. The paper tracks that error and claims a sharp scalar inequality absorbs it without weakening the target constant. Equality would then force the cone to decompose into independent half-lines, whose bounded section is a simplex. That is the proposed replacement for proving the minimum through dimension-specific vertex motions.

These are substantive mathematical commitments. The analytic estimates, their extension to arbitrary bodies and the equality classifications all need to hold. Our square and triangle calculations establish none of those steps; they show what the proposed machinery must explain.

The discovery was divided into separate attempts

OpenAI's reasoning summary separates three tasks: proving the symmetric bound, classifying its equality cases, and attacking the general-body problem.

The distinction changes the capability claim. The equality attempt takes a previously supplied sharp symmetric bound as a premise and reconstructs an earlier candidate argument. The summary does not identify that supplied premise with Part I's attempt. A later solution working from those inputs would demonstrate reasoning from provided mathematical material; the record does not establish one uninterrupted autonomous discovery of the whole theorem.

The reported search also shows routes that did not deliver the result. In the symmetric attempt, section-by-section induction and equal-measure partitions appear before the analytic boundary-law construction. In the general attempt, tensor powers, triangulation and a Cauchy–Schwarz comparison fail to provide the needed sharp argument. The later account develops an adaptive Gaussian comparison, then returns to the order of matrix operations, normalisation and equality.

We read that as an attempt to solve the same difficulty visible in the human history: a plausible lower estimate is insufficient if a loss prevents it reaching the exact constant. The proposed response is to redesign the comparison so those losses can be controlled. The summary exposes that direction of search; it is not a complete transcript or independent evidence that every proposed repair works.

What a field could build from a universal proof

If the proposed arguments hold, convex geometers would gain both an exact lower bound and an account of every shape that attains it. Other proofs could use the result. The classification would also make a further question natural. How closely must a nearly optimal shape resemble an exact minimiser? Settling the conjecture would end one search while giving the field a firmer basis for others.

That is why we care about the proposed method as much as the answer. The history shows how an argument can reach the exact constant in three dimensions yet leave the general problem open. A successful construction across dimensions would offer researchers a way to think about that obstacle. Its value to the community would grow if people could understand which parts of the comparison can be adapted to other questions.

There is a reason for scientists outside geometry to watch that development. Many scientific investigations require an argument about what a mathematical model implies, alongside evidence about whether the model describes the world. A system that can invent a useful argument in geometry is a reason to investigate its capabilities in those settings. It supplies no automatic evidence that the system can choose a sound physical model or design a decisive experiment.

Our position is that machine discovery should enlarge the research people can pursue. In this case, that would mean a result and an intelligible method that geometers can question, teach and extend. OpenAI's separate attempts and supplied premise still matter to the account of who contributed what. But the larger prospect is a field moving forward through a new source of ideas, with the next generation of researchers able to make those ideas their own.

Source note: we compared selected theorem statements and proof overviews at snapshot adc7f1241b42e322a6451854ab7e4b4c146bf78a with the literature linked above, and checked the polygon areas and vertex pairings directly. We also read the released reasoning summary. Sources checked 8 October 2026; we haven't independently certified the manuscripts or run a Lean build.