On a generalization of the Selection Theorem of Mahler

Gilbert Muraz[1]; Jean-Louis Verger-Gaugry[1]

  • [1] Institut Fourier - CNRS UMR 5582 Université de Grenoble I BP 74 - Domaine Universitaire 38402 Saint Martin d’Hères, France

Journal de Théorie des Nombres de Bordeaux (2005)

  • Volume: 17, Issue: 1, page 237-269
  • ISSN: 1246-7405

Abstract

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The set 𝒰 𝒟 r of point sets of ℝ n , n ≥ 1 , having the property that their minimal interpoint distance is greater than a given strictly positive constant r > 0 is shown to be equippable by a metric for which it is a compact topological space and such that the Hausdorff metric on the subset 𝒰 𝒟 r , f ⊂ 𝒰 𝒟 r of the finite point sets is compatible with the restriction of this topology to 𝒰 𝒟 r , f . We show that its subsets of Delone sets of given constants in ℝ n , n ≥ 1 , are compact. Three (classes of) metrics, whose one of crystallographic nature, requiring a base point in the ambient space, are given with their corresponding properties, for which we show topological equivalence. The point-removal process is proved to be uniformly continuous at infinity. We prove that this compactness Theorem implies the classical Selection Theorem of Mahler. We discuss generalizations of this result to ambient spaces other than ℝ n . The space 𝒰 𝒟 r is the space of equal sphere packings of radius r / 2 .

How to cite

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Muraz, Gilbert, and Verger-Gaugry, Jean-Louis. "On a generalization of the Selection Theorem of Mahler." Journal de Théorie des Nombres de Bordeaux 17.1 (2005): 237-269. <http://eudml.org/doc/249457>.

@article{Muraz2005,
abstract = {The set $\mathcal\{U\}\mathcal\{D\}_\{r\}$ of point sets of $\mathbb\{R\}^\{n\}, n \ge 1$, having the property that their minimal interpoint distance is greater than a given strictly positive constant $r &gt; 0$ is shown to be equippable by a metric for which it is a compact topological space and such that the Hausdorff metric on the subset $\mathcal\{U\}\mathcal\{D\}_\{r,f\} \subset \mathcal\{U\}\mathcal\{D\}_\{r\}$ of the finite point sets is compatible with the restriction of this topology to $\mathcal\{U\}\mathcal\{D\}_\{r,f\}$. We show that its subsets of Delone sets of given constants in $\mathbb\{R\}^\{n\}, n \ge 1$, are compact. Three (classes of) metrics, whose one of crystallographic nature, requiring a base point in the ambient space, are given with their corresponding properties, for which we show topological equivalence. The point-removal process is proved to be uniformly continuous at infinity. We prove that this compactness Theorem implies the classical Selection Theorem of Mahler. We discuss generalizations of this result to ambient spaces other than $\mathbb\{R\}^\{n\}$. The space $\mathcal\{U\}\mathcal\{D\}_\{r\}$ is the space of equal sphere packings of radius $r/2$.},
affiliation = {Institut Fourier - CNRS UMR 5582 Université de Grenoble I BP 74 - Domaine Universitaire 38402 Saint Martin d’Hères, France; Institut Fourier - CNRS UMR 5582 Université de Grenoble I BP 74 - Domaine Universitaire 38402 Saint Martin d’Hères, France},
author = {Muraz, Gilbert, Verger-Gaugry, Jean-Louis},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Mahler's selection theorem; Delone sets; Hausdorff metric},
language = {eng},
number = {1},
pages = {237-269},
publisher = {Université Bordeaux 1},
title = {On a generalization of the Selection Theorem of Mahler},
url = {http://eudml.org/doc/249457},
volume = {17},
year = {2005},
}

TY - JOUR
AU - Muraz, Gilbert
AU - Verger-Gaugry, Jean-Louis
TI - On a generalization of the Selection Theorem of Mahler
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2005
PB - Université Bordeaux 1
VL - 17
IS - 1
SP - 237
EP - 269
AB - The set $\mathcal{U}\mathcal{D}_{r}$ of point sets of $\mathbb{R}^{n}, n \ge 1$, having the property that their minimal interpoint distance is greater than a given strictly positive constant $r &gt; 0$ is shown to be equippable by a metric for which it is a compact topological space and such that the Hausdorff metric on the subset $\mathcal{U}\mathcal{D}_{r,f} \subset \mathcal{U}\mathcal{D}_{r}$ of the finite point sets is compatible with the restriction of this topology to $\mathcal{U}\mathcal{D}_{r,f}$. We show that its subsets of Delone sets of given constants in $\mathbb{R}^{n}, n \ge 1$, are compact. Three (classes of) metrics, whose one of crystallographic nature, requiring a base point in the ambient space, are given with their corresponding properties, for which we show topological equivalence. The point-removal process is proved to be uniformly continuous at infinity. We prove that this compactness Theorem implies the classical Selection Theorem of Mahler. We discuss generalizations of this result to ambient spaces other than $\mathbb{R}^{n}$. The space $\mathcal{U}\mathcal{D}_{r}$ is the space of equal sphere packings of radius $r/2$.
LA - eng
KW - Mahler's selection theorem; Delone sets; Hausdorff metric
UR - http://eudml.org/doc/249457
ER -

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