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Decomposable polytopes with few vertices
Abstract
A polytope P is decomposable if it can be written as the Minkowski sum of two polytopes that are not homothetic to it. It is well known that decomposable d-dimensional polytopes with 2d or fewer vertices are the simplicial prisms. More recent work shows that, with two exceptions in low dimensions, decomposable d-dimensional polytopes with exactly 2d+1 vertices are either pentasms or capped prisms. All these examples have a line segment for a summand. In this paper, we extend this conclusion to decomposable d-polytopes with exactly 2d+2 or 2d+3 vertices. In both cases, all such polytopes, with three exceptions, have a line segment for a summand. The exceptional cases are specified in detail.
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BibTeX
@misc{wang2026decomposable,
title = {{Decomposable polytopes with few vertices}},
author = {Jie Wang and David Yost},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07472},
eprint = {2611.07472},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07472},
}
RIS
TY - UNPB
TI - Decomposable polytopes with few vertices
AU - Jie Wang
AU - David Yost
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.07472
UR - https://arxiv.org/abs/2611.07472
AB - A polytope P is decomposable if it can be written as the Minkowski sum of two polytopes that are not homothetic to it. It is well known that decomposable d-dimensional polytopes with 2d or fewer vertices are the simplicial prisms. More recent work shows that, with two exceptions in low dimensions, decomposable d-dimensional polytopes with exactly 2d+1 vertices are either pentasms or capped prisms. All these examples have a line segment for a summand. In this paper, we extend this conclusion to decomposable d-polytopes with exactly 2d+2 or 2d+3 vertices. In both cases, all such polytopes, with three exceptions, have a line segment for a summand. The exceptional cases are specified in detail.
ER -