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Stability of the simplex bound for packings by equal spherical caps determined by simplicial regular polytopes. (English) Zbl 1418.52005

Conder, Marston D. E. (ed.) et al., Discrete geometry and symmetry. Dedicated to Károly Bezdek and Egon Schulte on the occasion of their 60th birthdays. Selected papers based on the presentations at the conference ‘Geometry and symmetry’, Veszprém, Hungary, June 29 – July 3, 2015. Cham: Springer. Springer Proc. Math. Stat. 234, 31-60 (2018).
The paper under review investigates optimal packing of equal spherical balls associated to regular polytopes in Euclidean spaces. Let \(d \geq 3\) and write \(S^{d-1}\) for the unit sphere centered at the origin in \(\mathbb{R}^d\). Let \(P\) be a Euclidean simplicial regular polytope with vertices on \(S^{d-1}\): such polytopes are regular simplex and crosspolytope in all dimensions, as well as icosahedron in \(\mathbb{R}^3\) and the \(600\)-cell in \(\mathbb{R}^4\). Let \(\varphi_P\) denote the acute angle such that edge length of \(P\) is \(2 \sin \varphi_P\). It has been previously proved (by multiple authors in the cases of different such polytopes) that the vertices of \(P\) are centers of an optimal packing of equal spherical balls of radius \(\varphi_P\) on \(S^{d-1}\). Starting with an overview of these previous results, the current paper moves to its own main theorem, a certain stability version of the above result.
With the same notation as above and suitable \(\varepsilon_P, c_P > 0\), let \(x_1,\dots,x_k \in S^{d-1}\) be centers of non-overlapping spherical balls of radius at least \(\varphi_P-\varepsilon\) for \(\varepsilon \in [0,\varepsilon_P)\) and \(k\) at least the number of vertices of \(P\). Then the authors prove that \(k\) is equal to the number of vertices of \(P\), and there exists a \(\Phi = O(d)\) such that for any \(x_i\) there is a vertex \(v\) of \(P\) with \(\delta(x_i,\Phi v) \leq c_P \varepsilon\), where \(\delta\) stands for the spherical distance, which is just the angle between the two corresponding vectors in \(\mathbb{R}^d\). The authors provide explicit expressions for \(\varepsilon_P\) and \(c_P\) for different polytopes. The proof techniques are also different for different polytopes: only linear algebra is needed for the case of a simplex, linear programming bound is used for the crosspolytopes, while icosahedron and the \(600\)-cell require the simplex bound.
For the entire collection see [Zbl 1400.52002].

MSC:

52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry)
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