On prescribing total preorders and linear orders to pairwise distances of points in Euclidean space
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866910014676926464 |
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| author | Almendra-Hernández, Víctor Hugo Martínez-Sandoval, Leonardo |
| author_facet | Almendra-Hernández, Víctor Hugo Martínez-Sandoval, Leonardo |
| contents | We show that any total preorder on a set with $\binom{n}{2}$ elements coincides with the order on pairwise distances of some point collection of size $n$ in $\mathbb{R}^{n-1}$. For linear orders, a collection of $n$ points in $\mathbb{R}^{n-2}$ suffices. These bounds turn out to be optimal. We also find an optimal bound in a bipartite version for total preorders and a near-optimal bound for a bipartite version for linear orders. Our arguments include tools from convexity and positive semidefinite quadratic forms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_08895 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On prescribing total preorders and linear orders to pairwise distances of points in Euclidean space Almendra-Hernández, Víctor Hugo Martínez-Sandoval, Leonardo Combinatorics Discrete Mathematics Metric Geometry 52C10 (Primary), 06A05 (Secondary) We show that any total preorder on a set with $\binom{n}{2}$ elements coincides with the order on pairwise distances of some point collection of size $n$ in $\mathbb{R}^{n-1}$. For linear orders, a collection of $n$ points in $\mathbb{R}^{n-2}$ suffices. These bounds turn out to be optimal. We also find an optimal bound in a bipartite version for total preorders and a near-optimal bound for a bipartite version for linear orders. Our arguments include tools from convexity and positive semidefinite quadratic forms. |
| title | On prescribing total preorders and linear orders to pairwise distances of points in Euclidean space |
| topic | Combinatorics Discrete Mathematics Metric Geometry 52C10 (Primary), 06A05 (Secondary) |
| url | https://arxiv.org/abs/2111.08895 |