Kissing polytopes in dimension 3
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866913709909082112 |
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| author | Deza, Antoine Liu, Zhongyuan Pournin, Lionel |
| author_facet | Deza, Antoine Liu, Zhongyuan Pournin, Lionel |
| contents | It is shown that the smallest possible distance between two disjoint lattice polytopes contained in the cube $[0,k]^3$ is exactly $$ \frac{1}{\sqrt{2(2k^2-4k+5)(2k^2-2k+1)}} $$ for every integer $k$ at least $4$. The proof relies on modeling this as a minimization problem over a subset of the lattice points in the hypercube $[-k,k]^9$. A precise characterization of this subset allows to reduce the problem to computing the roots of a finite number of degree at most $4$ polynomials, which is done using symbolic computation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_19554 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Kissing polytopes in dimension 3 Deza, Antoine Liu, Zhongyuan Pournin, Lionel Metric Geometry Combinatorics It is shown that the smallest possible distance between two disjoint lattice polytopes contained in the cube $[0,k]^3$ is exactly $$ \frac{1}{\sqrt{2(2k^2-4k+5)(2k^2-2k+1)}} $$ for every integer $k$ at least $4$. The proof relies on modeling this as a minimization problem over a subset of the lattice points in the hypercube $[-k,k]^9$. A precise characterization of this subset allows to reduce the problem to computing the roots of a finite number of degree at most $4$ polynomials, which is done using symbolic computation. |
| title | Kissing polytopes in dimension 3 |
| topic | Metric Geometry Combinatorics |
| url | https://arxiv.org/abs/2502.19554 |