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arXiv · 2502.19554

Kissing polytopes in dimension 3

Antoine Deza·Zhongyuan Liu·Lionel Pournin
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Abstract

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.

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Antoine Deza, Zhongyuan Liu, Lionel Pournin. 2025-02-26. Kissing polytopes in dimension 3. https://arxiv.org/abs/2502.19554

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