PreprintCC BY-NC-SA 4.0arXiv
Inscribed 4-polytopes of maximal volume with seven and eight vertices
Abstract
We prove, with sum-of-squares certificates checked in exact arithmetic, that every cyclic $4$-polytope with seven vertices inscribed in 𝕊3 has volume at most 2/5. Hence no $4$-polytope of maximal volume with seven vertices inscribed in 𝕊3 is cyclic, answering a question of Horváth and Lángi for d = 4, and with their theorem the maximal volume is √3/4. With eight vertices, the same method bounds the volume of every simplicial type other than the cross-polytope by 13/20, so the regular cross-polytope, of volume 2/3, is up to orthogonal transformations the unique maximiser.
Related
BibTeX
@misc{medgyes2026inscribed,
title = {{Inscribed 4-polytopes of maximal volume with seven and eight vertices}},
author = {Csaba Medgyes},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26886},
eprint = {2610.26886},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26886},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Inscribed 4-polytopes of maximal volume with seven and eight vertices
AU - Csaba Medgyes
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.26886
UR - https://arxiv.org/abs/2610.26886
AB - We prove, with sum-of-squares certificates checked in exact arithmetic, that every cyclic $4$-polytope with seven vertices inscribed in 𝕊^3 has volume at most 2/5. Hence no $4$-polytope of maximal volume with seven vertices inscribed in 𝕊^3 is cyclic, answering a question of Horváth and Lángi for d = 4, and with their theorem the maximal volume is √3/4. With eight vertices, the same method bounds the volume of every simplicial type other than the cross-polytope by 13/20, so the regular cross-polytope, of volume 2/3, is up to orthogonal transformations the unique maximiser.
ER -