Many equiprojective polytopes
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866908573883170816 |
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| author | Buffière, Théophile Pournin, Lionel |
| author_facet | Buffière, Théophile Pournin, Lionel |
| contents | A $3$-dimensional polytope $P$ is $k$-equiprojective when the projection of $P$ along any line that is not parallel to a facet of $P$ is a polygon with $k$ vertices. In 1968, Geoffrey Shephard asked for a description of all equiprojective polytopes. It has been shown recently that the number of combinatorial types of $k$-equiprojective polytopes is at least linear as a function of $k$. Here, it is shown that there are at least $k^{3k/2+o(k)}$ such combinatorial types as $k$ goes to infinity. This relies on the Goodman--Pollack lower bound on the number of order types and on new constructions of equiprojective polytopes via Minkowski sums. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_11366 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Many equiprojective polytopes Buffière, Théophile Pournin, Lionel Metric Geometry Combinatorics A $3$-dimensional polytope $P$ is $k$-equiprojective when the projection of $P$ along any line that is not parallel to a facet of $P$ is a polygon with $k$ vertices. In 1968, Geoffrey Shephard asked for a description of all equiprojective polytopes. It has been shown recently that the number of combinatorial types of $k$-equiprojective polytopes is at least linear as a function of $k$. Here, it is shown that there are at least $k^{3k/2+o(k)}$ such combinatorial types as $k$ goes to infinity. This relies on the Goodman--Pollack lower bound on the number of order types and on new constructions of equiprojective polytopes via Minkowski sums. |
| title | Many equiprojective polytopes |
| topic | Metric Geometry Combinatorics |
| url | https://arxiv.org/abs/2307.11366 |