PreprintCC BY 4.0arXiv
Lattice coverings by parallelotopes containing cube vertices
Abstract
We investigate when a parallelotope containing vertices of the unit cube Qn=[0,1]n covers Bbb Rn by integer translates. We prove the covering property for every parallelepiped containing five vertices of Q3, and, in any dimension, for every parallelotope containing a facet of Qn and one vertex of the opposite facet. As a consequence, we obtain the sharp bound 1/n for the volume of a pyramid contained in Qn with a parallelotope base. We also give an alternative proof of this bound.
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BibTeX
@misc{grozev2026lattice,
title = {{Lattice coverings by parallelotopes containing cube vertices}},
author = {Dragomir Grozev and Nikolai Nikolov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21572},
eprint = {2610.21572},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21572},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Lattice coverings by parallelotopes containing cube vertices
AU - Dragomir Grozev
AU - Nikolai Nikolov
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21572
UR - https://arxiv.org/abs/2610.21572
AB - We investigate when a parallelotope containing vertices of the unit cube Q_n=[0,1]^n covers Bbb R^n by integer translates. We prove the covering property for every parallelepiped containing five vertices of Q_3, and, in any dimension, for every parallelotope containing a facet of Q_n and one vertex of the opposite facet. As a consequence, we obtain the sharp bound 1/n for the volume of a pyramid contained in Q_n with a parallelotope base. We also give an alternative proof of this bound.
ER -