Minimal Covering Bodies and a Minkowski-Type Criterion for Lattice Coverings

Abstract

We study convex bodies whose translates by a fixed lattice cover space and for which every proper convex subbody loses this property. In three dimensions, the convex hull of independent translates of the six Kuhn tetrahedra always gives a lattice covering. We give a geometric proof using auxiliary tetrahedra and the parity of the covering multiplicity. We also prove polytopality and boundary restrictions for minimal covering bodies, and show that every three-dimensional parallelohedron admits a Kuhn representation with respect to its face-to-face tiling lattice. Constructions from Reeve tetrahedra give non-symmetric minimal covering bodies with eight vertices in dimension three and centrally symmetric ones with sixteen vertices in dimension four, with unbounded volumes for the integer covering lattice. They have pairwise distinct arithmetic contact types, which record lattice contacts and the faces containing them. The covering theorem also gives a finite intersection criterion for a prescribed lattice basis, with three intersection tests in the centrally symmetric case.

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BibTeX RIS

BibTeX

@misc{lian2026minimal,
  title = {{Minimal Covering Bodies and a Minkowski-Type Criterion for Lattice Coverings}},
  author = {Yanlu Lian and Fei Xue},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2606.14584},
  eprint = {2606.14584},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.14584},
}

RIS

TY  - UNPB
TI  - Minimal Covering Bodies and a Minkowski-Type Criterion for Lattice Coverings
AU  - Yanlu Lian
AU  - Fei Xue
PY  - 2026
DA  - 2026/09/13
PB  - arXiv
DO  - 10.48550/arxiv.2606.14584
UR  - https://arxiv.org/abs/2606.14584
AB  - We study convex bodies whose translates by a fixed lattice cover space and for which every proper convex subbody loses this property. In three dimensions, the convex hull of independent translates of the six Kuhn tetrahedra always gives a lattice covering. We give a geometric proof using auxiliary tetrahedra and the parity of the covering multiplicity. We also prove polytopality and boundary restrictions for minimal covering bodies, and show that every three-dimensional parallelohedron admits a Kuhn representation with respect to its face-to-face tiling lattice. Constructions from Reeve tetrahedra give non-symmetric minimal covering bodies with eight vertices in dimension three and centrally symmetric ones with sixteen vertices in dimension four, with unbounded volumes for the integer covering lattice. They have pairwise distinct arithmetic contact types, which record lattice contacts and the faces containing them. The covering theorem also gives a finite intersection criterion for a prescribed lattice basis, with three intersection tests in the centrally symmetric case.
ER  -

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