Empty simplices of large width
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2021
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912217909166080 |
|---|---|
| author | Doolittle, Joseph Katthän, Lukas Nill, Benjamin Santos, Francisco |
| author_facet | Doolittle, Joseph Katthän, Lukas Nill, Benjamin Santos, Francisco |
| contents | An empty simplex is a lattice simplex in which vertices are the only lattice points. We show two constructions leading to the first known empty simplices of width larger than their dimension:
- We introduce cyclotomic simplices and exhaustively compute all the cyclotomic simplices of dimension $10$ and volume up to $2^{31}$. Among them we find five empty ones of width $11$, and none of larger width.
- Using circulant matrices of a very specific form, we construct empty simplices of arbitrary dimension $d$ and width growing asymptotically as $d/\operatorname{arcsinh}(1) \sim 1.1346\,d$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2103_14925 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Empty simplices of large width Doolittle, Joseph Katthän, Lukas Nill, Benjamin Santos, Francisco Combinatorics Metric Geometry 52B20, 52C07, 52B15 An empty simplex is a lattice simplex in which vertices are the only lattice points. We show two constructions leading to the first known empty simplices of width larger than their dimension: - We introduce cyclotomic simplices and exhaustively compute all the cyclotomic simplices of dimension $10$ and volume up to $2^{31}$. Among them we find five empty ones of width $11$, and none of larger width. - Using circulant matrices of a very specific form, we construct empty simplices of arbitrary dimension $d$ and width growing asymptotically as $d/\operatorname{arcsinh}(1) \sim 1.1346\,d$. |
| title | Empty simplices of large width |
| topic | Combinatorics Metric Geometry 52B20, 52C07, 52B15 |
| url | https://arxiv.org/abs/2103.14925 |