Minimum Perimeter of Central Hyperplane Sections of the Cube
Manuscript 5 October 2026 · Online 5 October 2026
Abstract
For every integer \(n\ge3\), a central hyperplane section of the side-one real cube satisfies
\[\mathcal H^{n-2}\!\left(\partial_{u^\perp}([-1/2,1/2]^n\cap u^\perp)\right)\ge2(n-1).\]
Equality holds exactly for coordinate sections. The proof is analytic in dimensions three and four and every dimension at least seven; dimensions five and six use complete finite certificates checked with exact arithmetic.
Ordered facet densities transfer a cubic negative-moment estimate for section volume to perimeter, while a quantitative comparison of nearby graph sections validates every cell of the finite certificates. This answers the minimum-perimeter subquestion, item 2 of AIM Problem 1, not the other questions in the bundled dataset record.
Record
- Affiliation
- Mercury Software GmbH
- Contact
- [email protected] · GitHub
- Result
- Minimum-perimeter theorem
- Categories
- math.MG · math.FA
- Manuscript
- 5 October 2026
- Online release
- 5 October 2026
- Version
- 1.0
- License
- Creative Commons Attribution 4.0 International
Files and verification
The PDF is the canonical reading copy. The source archive contains the LaTeX manuscript, bibliography, reproducibility material, and audit documents without build artefacts.
Citation
Alper Ferudun, “Minimum Perimeter of Central Hyperplane Sections of the Cube,” EulerSolve Research Papers, AIM-CONVEX_GEOMETRY-0001, 2026. https://doi.org/10.5281/zenodo.23154580.
BibTeX
@misc{Ferudun2026CubeMinimumPerimeter,
author = {Ferudun, Alper},
title = {Minimum Perimeter of Central Hyperplane Sections of the Cube},
year = {2026},
howpublished = {EulerSolve Research Papers},
url = {https://eulersolve.org/papers/aim-convex_geometry-0001/},
doi = {10.5281/zenodo.23154580},
note = {AIM-CONVEX_GEOMETRY-0001; unrefereed preprint}
}