Cutting convex polytopes by hyperplanes
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mathematics-07-00381.pdf
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269.89 KB
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Adobe PDF
Checksum (MD5)
9dc259056a3864659649b7cd547baf6f
Author(s)
Hibi, Takayuki
Li, Nan
Date Issued
April 26, 2019
Journal
Mathematics
Publisher
Multidisciplinary Digital Publishing Institute
Citation
Hibi, Takayuki and Nan Li, "Cutting convex polytopes by hyperplanes." Mathematics 7, 5 (Apr. 2019): no. 381 doi 10.3390/math7050381 ©2019 Author(s)
Version
Final published version
Abstract
Cutting a polytope is a very natural way to produce new classes of interesting polytopes. Moreover, it has been very enlightening to explore which algebraic and combinatorial properties of the original polytope are hereditary to its subpolytopes obtained by a cut. In this work, we devote our attention to all the separating hyperplanes for some given polytope (integral and convex) and study the existence and classification of such hyperplanes. We prove the existence of separating hyperplanes for the order and chain polytopes for any finite posets that are not a single chain, and prove there are no such hyperplanes for any Birkhoff polytopes. Moreover, we give a complete separating hyperplane classification for the unit cube and its subpolytopes obtained by one cut, together with some partial classification results for order and chain polytopes. Keywords: separating hyperplane; order polytopes; chain polytopes; Birkhoff polytopes
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution
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DOI of Published Version
https://doi.org/10.3390/math7050381
