Embedding Stacked Polytopes on a Polynomial-Size Grid
A stacking operation adds a d-simplex on top of a facet of a simplicial d-polytope while maintaining the convexity of the polytope. A stacked d-polytope is a polytope that is obtained from a d-simplex and a series of stacking operations. We show that for a fixed d every stacked d-polytope with n ver...
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| Muut tekijät: | |
| Aineistotyyppi: | Artikkeli |
| Kieli: | English |
| Julkaistu: |
Springer US
2017
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| Linkit: | http://hdl.handle.net/1721.1/108786 https://orcid.org/0000-0003-3803-5703 |
| Yhteenveto: | A stacking operation adds a d-simplex on top of a facet of a simplicial d-polytope while maintaining the convexity of the polytope. A stacked d-polytope is a polytope that is obtained from a d-simplex and a series of stacking operations. We show that for a fixed d every stacked d-polytope with n vertices can be realized with nonnegative integer coordinates. The coordinates are bounded by O(n[superscript 2 log[subscript 2](2d)], except for one axis, where the coordinates are bounded by O(n[superscript 3 log[subscript 2](2d)]. The described realization can be computed with an easy algorithm. The realization of the polytopes is obtained with a lifting technique which produces an embedding on a large grid. We establish a rounding scheme that places the vertices on a sparser grid, while maintaining the convexity of the embedding. |
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