An example of an "unlinked" set of $2k + 3$ points in $2k$-space
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913234363088896 |
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| author | Starkov, M. |
| author_facet | Starkov, M. |
| contents | Take any $d + 3$ points in $\mathbb{R}^d$. It is known that
(a) if $d = 2k + 1$, then there are two linked $(k + 1)$-simplices with the vertices at these points;
(b) if $d = 2k$, then there are two disjoint $(k + 1)$-tuples of these points such that their convex hulls intersect.
The analogue of (b) for $d = 2k + 1$, which is also the analogue of (a) for intersections (instead of linkings), states that there are two disjoint $(k + 1)$- and $(k + 2)$-tuples of these points such that their convex hulls intersect. This analogue is correct by (a). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_09002 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An example of an "unlinked" set of $2k + 3$ points in $2k$-space Starkov, M. Combinatorics Metric Geometry 52C35, 52A20, 52A37 Take any $d + 3$ points in $\mathbb{R}^d$. It is known that (a) if $d = 2k + 1$, then there are two linked $(k + 1)$-simplices with the vertices at these points; (b) if $d = 2k$, then there are two disjoint $(k + 1)$-tuples of these points such that their convex hulls intersect. The analogue of (b) for $d = 2k + 1$, which is also the analogue of (a) for intersections (instead of linkings), states that there are two disjoint $(k + 1)$- and $(k + 2)$-tuples of these points such that their convex hulls intersect. This analogue is correct by (a). |
| title | An example of an "unlinked" set of $2k + 3$ points in $2k$-space |
| topic | Combinatorics Metric Geometry 52C35, 52A20, 52A37 |
| url | https://arxiv.org/abs/2402.09002 |