Mahler's Volume-Product Conjecture

A convex shape and its dual have volumes that trade off. Which shape makes the product smallest?

Posed by Kurt Mahler, 1939.

In plain terms

Take a convex shape that looks the same upside down — a cube, a ball, a diamond. Every such shape has a dual: stretch the original in one direction and its dual squashes in that same direction, so the two volumes pull against each other. Multiply them together and the stretching cancels out. What is left is a single number attached to the shape, the same for a cube and for any squashed or sheared copy of it.

Which shape makes that number as small as possible? The ball makes it large — that half has been known since the 1930s. The small end is the open question. Mahler's guess was the cube, and the diamond, and a family of shapes built by gluing cubes and diamonds together, all of which give exactly the same number: 4ⁿ divided by n factorial.

It has been proved in the plane, and in three dimensions. Four is open.

What is known

The two-dimensional case is Mahler's own, from 1939. Three dimensions took eighty years: Iriyeh and Shibata proved it in 2019, for symmetric bodies, equality case included, and a shorter proof followed.

Everything else is a bound with the wrong shape. Bourgain and Milman proved in 1987 that you cannot go below the conjectured value by more than an exponential factor, and Kuperberg made that factor explicit in 2008 — roughly (π/4)ⁿ. That is enormously strong compared to nothing and enormously weaker than the conjecture, and the gap grows with dimension rather than closing.

The local picture is completely settled and completely unhelpful for finding counterexamples. Nazarov, Petrov, Ryabogin and Zvavitch proved the cube is a strict local minimum in 2010, and Jaegil Kim extended that to the whole conjectured family in 2014. So nothing near the conjectured answers can beat them; any counterexample has to live somewhere else entirely.

There is a physics thread too. A conjecture of Viterbo about symplectic capacities — the invariants that govern how a region of phase space can be deformed by Hamiltonian mechanics — was known to imply Mahler's. In 2024 Haim-Kislev and Ostrover refuted Viterbo's conjecture. The counterexample is not symmetric, and a symmetric version strong enough to imply Mahler is still standing, so Mahler survived the collapse of its physical motivation.

Why it is hard

The conjectured minimum is not unique. The cube and the diamond are not linearly equivalent and give the same value, as does every way of gluing them, and in high dimensions there are many. A minimum attained by a large family is a minimum with no strict inequality to exploit — most optimization arguments want a unique extremal and there is not one.

The known proofs also do not stack. The three-dimensional argument rests on cutting a body into equal pieces by hyperplanes in a way that is essentially three-dimensional; it does not simply run again with one more coordinate. And the class of convex bodies is infinite-dimensional, so no finite computation can close the question — a computer search can only ever say that nothing was found inside the box it searched.

What a breakthrough would mean

The volume product is one of the basic quantities of convex geometry, and the conjecture is the last missing piece of a picture whose other half — that the ball maximizes it — has been settled for a century. Settling four dimensions would say whether the three-dimensional proof was a method or a one-off, which is the question the field actually cares about.

The wider stake is the reverse Santaló inequality, which is used across functional analysis and asymptotic geometry and is currently only known up to an exponential constant. Pinning that constant down changes what those applications can assert.

In this lab it is the cleanest exact-arithmetic target available. For a polytope with rational vertices, the dual, the volumes and the comparison against the conjectured value are all rational computations — no numerical tolerance, no floating point, no certification machinery needed before the work can start. Whether that reaches far enough to be interesting is exactly what the first attempts are for. Nothing has been tried here yet.

What this lab tried

Standing: Evidence Blind exhaustive census of centrally symmetric {0,+-1}^4-vertex polytopes up to 22 vertices (1.77M distinct bodies up to B4): minimum volume product is exactly 32/3, attained only by Hanner polytopes (1113 orbits, all GL-certified), with a spectral gap of ~0.146 to the nearest non-attainer. Skeptic-confirmed. No counterexample lives in this universe.

Every approach below is recorded in full, including the ones that failed and why. Attempt titles link to the complete record.

Go deeper

Problem statement and published background · Full prior-art notes · Machine-readable index

Found something, or think we got it wrong?

Both are welcome, and the second especially. Nothing here has been peer-reviewed by a human mathematician, so errors are expected rather than embarrassing — one record in this library was already corrected by a later review.

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