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 |
| Language: | English |
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MDPI AG
2019-04-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/7/5/381 |
| _version_ | 1828963376422715392 |
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| author | Takayuki Hibi Nan Li |
| author_facet | Takayuki Hibi Nan Li |
| author_sort | Takayuki Hibi |
| collection | DOAJ |
| 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. |
| first_indexed | 2024-12-14T10:26:15Z |
| format | Article |
| id | doaj.art-7a3f8d18a0184b8bb1e304cac905a79d |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-12-14T10:26:15Z |
| publishDate | 2019-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-7a3f8d18a0184b8bb1e304cac905a79d2022-12-21T23:06:19ZengMDPI AGMathematics2227-73902019-04-017538110.3390/math7050381math7050381Cutting Convex Polytopes by HyperplanesTakayuki Hibi0Nan Li1Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Toyonaka, Osaka 560-0043, JapanDepartment of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USACutting 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.https://www.mdpi.com/2227-7390/7/5/381separating hyperplaneorder polytopeschain polytopesBirkhoff polytopes |
| spellingShingle | Takayuki Hibi Nan Li Cutting Convex Polytopes by Hyperplanes Mathematics separating hyperplane order polytopes chain polytopes Birkhoff polytopes |
| 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 |
| topic | separating hyperplane order polytopes chain polytopes Birkhoff polytopes |
| url | https://www.mdpi.com/2227-7390/7/5/381 |
| work_keys_str_mv | AT takayukihibi cuttingconvexpolytopesbyhyperplanes AT nanli cuttingconvexpolytopesbyhyperplanes |