Cutting convex polytopes by hyperplanes
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 attenti...
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| Format: | Article |
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Multidisciplinary Digital Publishing Institute
2020
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| Online Access: | https://hdl.handle.net/1721.1/125362 |
| _version_ | 1811080758937255936 |
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| author | Hibi, Takayuki Li, Nan |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Hibi, Takayuki Li, Nan |
| author_sort | Hibi, Takayuki |
| collection | MIT |
| description | 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 |
| first_indexed | 2024-09-23T11:36:23Z |
| format | Article |
| id | mit-1721.1/125362 |
| institution | Massachusetts Institute of Technology |
| last_indexed | 2024-09-23T11:36:23Z |
| publishDate | 2020 |
| publisher | Multidisciplinary Digital Publishing Institute |
| record_format | dspace |
| spelling | mit-1721.1/1253622022-10-01T04:46:45Z Cutting convex polytopes by hyperplanes Hibi, Takayuki Li, Nan Massachusetts Institute of Technology. Department of Mathematics 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 2020-05-20T20:14:43Z 2020-05-20T20:14:43Z 2019-04-26 2019-02 2020-03-02T12:51:36Z Article http://purl.org/eprint/type/JournalArticle 2227-7390 https://hdl.handle.net/1721.1/125362 Hibi, Takayuki and Nan Li, "Cutting convex polytopes by hyperplanes." Mathematics 7, 5 (Apr. 2019): no. 381 doi 10.3390/math7050381 ©2019 Author(s) 10.3390/math7050381 Mathematics Creative Commons Attribution https://creativecommons.org/licenses/by/4.0/ application/pdf Multidisciplinary Digital Publishing Institute Multidisciplinary Digital Publishing Institute |
| spellingShingle | Hibi, Takayuki Li, Nan Cutting convex polytopes by hyperplanes |
| title | Cutting convex polytopes by hyperplanes |
| title_full | Cutting convex polytopes by hyperplanes |
| title_fullStr | Cutting convex polytopes by hyperplanes |
| title_full_unstemmed | Cutting convex polytopes by hyperplanes |
| title_short | Cutting convex polytopes by hyperplanes |
| title_sort | cutting convex polytopes by hyperplanes |
| url | https://hdl.handle.net/1721.1/125362 |
| work_keys_str_mv | AT hibitakayuki cuttingconvexpolytopesbyhyperplanes AT linan cuttingconvexpolytopesbyhyperplanes |