0/1-polytope: Difference between revisions
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{{Short description|Type of convex polytope}} |
{{Short description|Type of convex polytope}} |
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| ⚫ | A '''0/1-polytope''' is a [[convex polytope]] generated by the [[convex hull]] of a subset of {{math|1=''d''}} coordinates value 0 or 1, {{math|1={0,1}<sup>''d''</sup>}}.{{r|ziegler}} The full domain is the unit [[hypercube]] with cut [[hyperplane]]s passing through these coordinates.{{r|grunbaum}} A {{mvar|d}}-polytope requires at least {{math|''d'' + 1}} vertices, and can't be all in the same hyperplanes. |
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{{one source |date=May 2024}} |
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| ⚫ | A '''0/1-polytope''' is a [[convex polytope]] generated by the [[convex hull]] of a subset of {{math|1=''d''}} coordinates value 0 or 1, {{math|1={0,1}<sup>''d''</sup>}}. The full domain is the unit [[hypercube]] with cut [[hyperplane]]s passing through these coordinates.{{r|grunbaum}} A {{mvar|d}}-polytope requires at least {{math|''d'' + 1}} vertices, and can't be all in the same hyperplanes. |
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{{nowrap|1=''n''-}}[[simplex]] polytopes for example can be generated {{math|1=''n'' + 1}} vertices, using the origin, and one vertex along each primary axis, {{math|1=(1,0....)}}, etc. |
{{nowrap|1=''n''-}}[[simplex]] polytopes for example can be generated {{math|1=''n'' + 1}} vertices, using the origin, and one vertex along each primary axis, {{math|1=(1,0....)}}, etc. Every [[simple polytope|simple]] 0/1-polytope is a [[Proprism|Cartesian product]] of 0/1 simplexes.{{r|kw}} |
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==References== |
==References== |
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{{reflist|refs= |
{{reflist|refs= |
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<ref name= |
<ref name=grunbaum>{{cite book |
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| last = Grünbaum | first = Branko | authorlink = Branko Grünbaum |
| last = Grünbaum | first = Branko | authorlink = Branko Grünbaum |
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| title = Convex Polytopes |
| title = Convex Polytopes |
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| url = https://books.google.com/books?id=5iV75P9gIUgC&pg=PA69 |
| url = https://books.google.com/books?id=5iV75P9gIUgC&pg=PA69 |
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}}</ref> |
}}</ref> |
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<ref name=kw>{{citation |
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| last1 = Kaibel | first1 = Volker |
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| last2 = Wolff | first2 = Martin |
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| doi = 10.1006/eujc.1999.0328 |
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| issue = 1 |
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| journal = European Journal of Combinatorics |
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| mr = 1737334 |
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| pages = 139–144 |
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| title = Simple 0/1-polytopes |
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| volume = 21 |
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| year = 2000}}</ref> |
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<ref name=ziegler>{{citation |
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| last = Ziegler | first = Günter M. | author-link = Günter Ziegler |
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| contribution = Lectures on 0/1-polytopes |
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| isbn = 3-7643-6351-7 |
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| mr = 1785291 |
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| pages = 1–41 |
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| publisher = Birkhäuser | location = Basel |
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| series = DMV Sem. |
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| title = Polytopes—combinatorics and computation (Oberwolfach, 1997) |
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| volume = 29 |
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| year = 2000}}</ref> |
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}} |
}} |
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Revision as of 00:40, 15 May 2024
A 0/1-polytope is a convex polytope generated by the convex hull of a subset of d coordinates value 0 or 1, {0,1}d.[1] The full domain is the unit hypercube with cut hyperplanes passing through these coordinates.[2] A d-polytope requires at least d + 1 vertices, and can't be all in the same hyperplanes.
n-simplex polytopes for example can be generated n + 1 vertices, using the origin, and one vertex along each primary axis, (1,0....), etc. Every simple 0/1-polytope is a Cartesian product of 0/1 simplexes.[3]
References
- ^ Ziegler, Günter M. (2000), "Lectures on 0/1-polytopes", Polytopes—combinatorics and computation (Oberwolfach, 1997), DMV Sem., vol. 29, Basel: Birkhäuser, pp. 1–41, ISBN 3-7643-6351-7, MR 1785291
- ^ Grünbaum, Branko (2003). "4.9. Additional notes and comments". Convex Polytopes. Springer. p. 69a.
- ^ Kaibel, Volker; Wolff, Martin (2000), "Simple 0/1-polytopes", European Journal of Combinatorics, 21 (1): 139–144, doi:10.1006/eujc.1999.0328, MR 1737334
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