OP-00695 Solution claimed

The symmetric Mahler inequality in four dimensions

Let $K \subset \mathbb{R}^4$ be a convex body, that is, a compact convex set with nonempty interior, with $K = -K$, and let $$K^\circ = \{y \in \mathbb{R}^4 : \langle x, y \rangle \le 1 \text{ for all } x \in K\}$$ be its polar body. Mahler conjectured that every centrally symmetric convex body $K \subset \mathbb{R}^n$ satisfies $\operatorname{vol}_n(K) \operatorname{vol}_n(K^\circ) \ge 4^n/n!$, the value for the cube and its polar cross-polytope. He proved this for $n = 2$, and Iriyeh and Shibata proved it for $n = 3$, with equality exactly when $K$ or $K^\circ$ is a parallelepiped.

Problem. Is $$\operatorname{vol}_4(K) \operatorname{vol}_4(K^\circ) \ge \frac{4^4}{4!}$$ for every such $K$? No polytope, unconditional-symmetry or smoothness restriction is placed on $K$, and only the inequality is asked, not the equality cases.

Source: Mahler (1939), the symmetric Mahler conjecture in every dimension; stated for n = 4 by AIM — Open Applied Problems.

Imported from github.com/MColbrook/AIM.

Classification

Primary: MG — Metric Geometry
Secondary: FA
Keywords: AIM Open Applied Problems, convex bodies, Mahler conjecture, volume product

References

@article{mahler1939,
  author  = {Mahler, Kurt},
  title   = {Ein {\"U}bertragungsprinzip f{\"u}r konvexe {K}{\"o}rper},
  journal = {{\v C}asopis pro p{\v e}stov{\'a}n{\'i} matematiky a fysiky},
  volume  = {68},
  number  = {3},
  year    = {1939},
  pages   = {93--102},
  doi     = {10.21136/CPMF.1939.109441},
}
@article{iriyeh2020,
  author        = {Iriyeh, Hiroshi and Shibata, Masataka},
  title         = {Symmetric {M}ahler's conjecture for the volume product in the 3-dimensional case},
  journal       = {Duke Math. J.},
  volume        = {169},
  number        = {6},
  year          = {2020},
  pages         = {1077--1134},
  doi           = {10.1215/00127094-2019-0072},
  eprint        = {1706.01749},
  archivePrefix = {arXiv},
}
@misc{aim-mahler-symmetric-four-dimensions,
  author       = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
  title        = {{The symmetric Mahler inequality in four dimensions}},
  howpublished = {AIM --- Open Applied Problems, GitHub repository},
  year         = {2026},
  url          = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/086-mahler-symmetric-four-dimensions.md},
  note         = {Entry 086 at commit 1b7c7fd, 2026-10-13},
}

Resolution claims

The symmetric Mahler conjecture and its equality cases v1 GitHub ↗

OpenAI

We resolve the symmetric Mahler conjecture positively, including its equality classification. Every origin-symmetric convex body in $\R^n$ has volume product at least $4^n/n!$, with equality exactly for invertible linear images of Hanner polytopes.

Full resolution · not verified · fidelity not established · posted on GitHub 6 Oct 2026

Status history

When	Change	By	Reason
2026-10-16	Needs verification → Solution claimed	—	Recorded here: math @ 1581d4c, build/main.tex.

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-00695, "The symmetric Mahler inequality in four dimensions", clopen.solutions.