Small kissing polytopes
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866911203151839232 |
|---|---|
| author | Deza, Antoine Liu, Zhongyuan Pournin, Lionel |
| author_facet | Deza, Antoine Liu, Zhongyuan Pournin, Lionel |
| contents | A lattice $(d,k)$-polytope is the convex hull of a set of points in $\mathbb{R}^d$ whose coordinates are integers ranging between $0$ and $k$. We consider the smallest possible distance $\varepsilon(d,k)$ between two disjoint lattice $(d,k)$-polytopes. We propose an algebraic model for this distance and derive from it an explicit formula for $\varepsilon(2,k)$. Our model also allows for the computation of previously intractable values of $\varepsilon(d,k)$. In particular, we compute $\varepsilon(3,k)$ when $4\leq{k}\leq8$, $\varepsilon(4,k)$ when $2\leq{k}\leq3$, and $\varepsilon(6,1)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_03479 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Small kissing polytopes Deza, Antoine Liu, Zhongyuan Pournin, Lionel Combinatorics Metric Geometry Optimization and Control A lattice $(d,k)$-polytope is the convex hull of a set of points in $\mathbb{R}^d$ whose coordinates are integers ranging between $0$ and $k$. We consider the smallest possible distance $\varepsilon(d,k)$ between two disjoint lattice $(d,k)$-polytopes. We propose an algebraic model for this distance and derive from it an explicit formula for $\varepsilon(2,k)$. Our model also allows for the computation of previously intractable values of $\varepsilon(d,k)$. In particular, we compute $\varepsilon(3,k)$ when $4\leq{k}\leq8$, $\varepsilon(4,k)$ when $2\leq{k}\leq3$, and $\varepsilon(6,1)$. |
| title | Small kissing polytopes |
| topic | Combinatorics Metric Geometry Optimization and Control |
| url | https://arxiv.org/abs/2412.03479 |