A sharp minimum for cube-apex bodies in every dimension
Abstract
We determine the minimum volume product of centrally symmetric polytopes obtained by adding facet apices to a cube while fixing their total outward height. Heights and tangential coordinates vary subject to a product condition on the polar-cap denominators. In every dimension at least two, the minimum has an explicit formula, attained by concentrating the height at one opposite pair of corner apices. For positive height in dimensions at least three, every minimizing array has this structure. The product condition strictly extends coordinatewise quadratic constraints. We also obtain two bounds without that condition: a four-dimensional Mahler bound for a specified height range, and a correction measuring volume generated jointly by two apices. The latter certifies a continuous family outside the preceding sufficient conditions. The proof combines known cap volumes with a sharp aggregate height estimate and a double-pyramid construction. Lean~4 verifies the actual-volume minimum, the necessary equality structure and both extensions.
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BibTeX
@misc{shen2026sharp,
title = {{A sharp minimum for cube-apex bodies in every dimension}},
author = {Hanyue Shen and Pavel B. Dubovski},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11014},
eprint = {2610.11014},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11014},
}
RIS
TY - UNPB
TI - A sharp minimum for cube-apex bodies in every dimension
AU - Hanyue Shen
AU - Pavel B. Dubovski
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.11014
UR - https://arxiv.org/abs/2610.11014
AB - We determine the minimum volume product of centrally symmetric polytopes obtained by adding facet apices to a cube while fixing their total outward height. Heights and tangential coordinates vary subject to a product condition on the polar-cap denominators. In every dimension at least two, the minimum has an explicit formula, attained by concentrating the height at one opposite pair of corner apices. For positive height in dimensions at least three, every minimizing array has this structure. The product condition strictly extends coordinatewise quadratic constraints. We also obtain two bounds without that condition: a four-dimensional Mahler bound for a specified height range, and a correction measuring volume generated jointly by two apices. The latter certifies a continuous family outside the preceding sufficient conditions. The proof combines known cap volumes with a sharp aggregate height estimate and a double-pyramid construction. Lean~4 verifies the actual-volume minimum, the necessary equality structure and both extensions.
ER -