On a generalization of the Selection Theorem of Mahler

33 pagesInternational audienceThe set ~\uc\dc_{r}~ of point sets of ~\rb^{n}, n \geq 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 ~\uc\dc_{r,f} \subset \uc\dc_{r}~ of the finite point sets is compatible with the restriction of this topology to ~\uc\dc_{r,f}. We show that its subsets of Delone sets of given constants in ~\rb^{n}, n \geq 1, are compact. Three (classes of) metrics, requiring a base point in the ambient space, are given with their corresponding properties, for which we show that their topological equivalence occurs under some assumptions. 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 ~\rb^{n}. The space ~\uc\dc_{r}~ is the space of equal sphere packings of radius~$r/2$