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Math-2449 · mathematics · Geometry

Mahler’s Volume Conjecture

Status Open · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
For every convex body $K\subseteq\mathbb R^n$, is its Mahler volume bounded by

$$\bar s(K)=\lvert K\rvert\inf_{x\in K}\lvert(K-x)^\circ\rvert\geq\frac{(n+1)^{n+1}}{(n!)^2},$$

with equality for a simplex?

Formulation references: [S1], [S2].

Short English statement

For every convex body $K\subseteq\mathbb R^n$, is its Mahler volume bounded by

$$\bar s(K)=\lvert K\rvert\inf_{x\in K}\lvert(K-x)^\circ\rvert\geq\frac{(n+1)^{n+1}}{(n!)^2},$$

with equality for a simplex?

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2449-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2449

The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 087; 2026-09-22). Other linked papers contribute only their stated special cases; those weaker results alone would not resolve the complete question.

Mahler’s Volume Conjecture

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 087; 2026-09-22), the asserted result is that general convex bodies satisfy the simplex lower bound $(n+1)^{n+1}/(n!)^2$ for their centered volume product, with simplex equality; In manuscript 2 (family 087; 2026-09-22), the asserted result is that origin-symmetric convex bodies in dimension $n$ have volume product at least $4^n/n!$, with equality precisely for linear images of Hanner polytopes.

Mathematical statement of the claimed result

Manuscript 1.
For a full-dimensional convex body $K\subset\mathbb R^n$, let $s(K)$ be its Santaló point, the minimizer of $|(K-z)^{\circ}|$ over interior $z$, and define $P(K)=|K|\,|(K-s(K))^{\circ}|$. The claimed nonsymmetric Mahler bound is \[ P(K)\ge\frac{(n+1)^{n+1}}{(n!)^2}\quad(n\ge1), \] Equality characterizes simplices among these convex bodies. No central-symmetry hypothesis is imposed; the polar is taken after Santaló centering.
Manuscript 2.
In each dimension $n\ge1$, take any convex body $K\subset\mathbb R^n$ with $K=-K$. Define $K^{\circ}=\{y:x\cdot y\le1\text{ for all }x\in K\}$. The manuscript claims \[ |K|\,|K^{\circ}|\ge\frac{4^n}{n!},\qquad |K|\,|K^{\circ}|=\frac{4^n}{n!}\ \Longleftrightarrow\ K\text{ is a linear Hanner body}. \] Here Hanner bodies are built from one-dimensional symmetric intervals by successive $\ell_1$ and $\ell_{\infty}$ sums, and a linear Hanner body is an invertible linear image of one. The bound and the complete equality classification hold in every dimension.

Outline of the argument

For manuscript 1 (family 087; 2026-09-22), lifting the body to a cone converts polarity into paired Laplace integrals. Biased projection coordinates and an entropy inequality control their product, and change of variables returns the geometric lower bound and rigidity.

For manuscript 2 (family 087; 2026-09-22), a functional inequality is obtained from a conformal-lens construction and a real simplex identity. Feasible bases and a common equality witness identify the extremizers, while approximation transfers the sharp estimate from polytopes to arbitrary bodies.

Formal verification

For manuscript 1 (family 087; 2026-09-22), OpenAI lists Lean statements for this assertion: general convex bodies satisfy the simplex lower bound $(n+1)^{n+1}/(n!)^2$ for their centered volume product, with simplex equality.

The selected statement includes the clause used for this problem.

For manuscript 2 (family 087; 2026-09-22), OpenAI lists Lean statements for this assertion: origin-symmetric convex bodies in dimension $n$ have volume product at least $4^n/n!$, with equality precisely for linear images of Hanner polytopes.

The central assertion is within the documented formal scope; the additional applications retain their own scope limits.

The linked family Comparator checks are GeneralMahler.lean, MahlerConjecture.lean, SymmetricMahlerEquality.lean, SymmetricPolar.lean.

Their named declarations include OAI.GeneralMahler.general_mahler, OAI.SymmetricMahler.symmetric_mahler, OAI.SymmetricMahler.symmetric_mahler_equality, OAI.SymmetricPolar.symmetric_polar_main.

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

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