Minimum-Volume Central Slabs of the Regular Octahedron
Manuscript 5 October 2026 · Online 5 October 2026
Abstract
We determine the minimum volume of the intersection of a regular octahedron with a centered slab of every prescribed width, including all equality directions. For the octahedron with vertices \(\pm e_i\) and slab half-width \(t\), the minimizing normal is diagonal below a single threshold \(\tau=0.4740349423895888\ldots\) and is a coordinate normal above it, until the slab contains the whole body.
Put \(\kappa=3\sqrt{3}/8\). The threshold is the unique root in \((0,1/\sqrt{3})\) of \[(1+\kappa)\tau^2-3\tau+3-3\kappa=0.\] The analytic proof derives explicit chamber formulas, reduces the optimization to normals with two equal coordinates, and excludes interior extrema using sign-monotone derivatives and a polynomial resultant.
This answers only the three-dimensional minimum-direction part of AIM Fourier-Convex Problem 23(d); higher dimensions and maximum volumes are not addressed.
Record
- Affiliation
- Mercury Software GmbH
- Contact
- [email protected] · GitHub
- Result
- Octahedron minimum-volume 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-Volume Central Slabs of the Regular Octahedron,” EulerSolve Research Papers, AIM-CONVEX_GEOMETRY-0045, 2026. https://doi.org/10.5281/zenodo.23156791.
BibTeX
@misc{Ferudun2026OctahedronMinimumSlabs,
author = {Ferudun, Alper},
title = {Minimum-Volume Central Slabs of the Regular Octahedron},
year = {2026},
howpublished = {EulerSolve Research Papers},
url = {https://eulersolve.org/papers/aim-convex_geometry-0045/},
doi = {10.5281/zenodo.23156791},
note = {AIM-CONVEX_GEOMETRY-0045; unrefereed preprint}
}