Flat simplices and kissing polytopes
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866908750230585344 |
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| author | Deza, Antoine Pournin, Lionel |
| author_facet | Deza, Antoine Pournin, Lionel |
| contents | We consider how flat a lattice simplex contained in the hypercube $[0,k]^d$ can be. This question is related to the notion of kissing polytopes: two lattice polytopes contained in the hypercube $[0,k]^d$ are kissing when they are disjoint but their distance is as small as possible. We show that the smallest possible distance of a lattice point $P$ contained in the cube $[0,k]^3$ to a lattice triangle in the same cube that does not contain $P$ is $$ \frac{1}{\sqrt{3k^4-4k^3+4k^2-2k+1}} $$ when $k$ is at least $2$. We also improve the known lower bounds on the distance of kissing polytopes for $d$ at least $4$ and $k$ at least $2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_03183 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Flat simplices and kissing polytopes Deza, Antoine Pournin, Lionel Metric Geometry Combinatorics We consider how flat a lattice simplex contained in the hypercube $[0,k]^d$ can be. This question is related to the notion of kissing polytopes: two lattice polytopes contained in the hypercube $[0,k]^d$ are kissing when they are disjoint but their distance is as small as possible. We show that the smallest possible distance of a lattice point $P$ contained in the cube $[0,k]^3$ to a lattice triangle in the same cube that does not contain $P$ is $$ \frac{1}{\sqrt{3k^4-4k^3+4k^2-2k+1}} $$ when $k$ is at least $2$. We also improve the known lower bounds on the distance of kissing polytopes for $d$ at least $4$ and $k$ at least $2$. |
| title | Flat simplices and kissing polytopes |
| topic | Metric Geometry Combinatorics |
| url | https://arxiv.org/abs/2601.03183 |