Abstract.
Let μ be a non-negative number not greater than 1. Consider an arrangement \({\cal S}\) of (not necessarily congruent) spheres with positive homogenity in the n-dimensional Euclidean space, i.e., in which the infimum of the radii of the spheres divided by the supremum of the radii of the spheres is a positive number. With each sphere S of \({\cal S}\) associate a concentric sphere of radius μ times the radius of S. We call this sphere the μ-kernel of S. The arrangement \({\cal S}\) is said to be a Minkowski arrangement of order μ if no sphere of \({\cal S}\) overlaps the μ-kernel of another sphere. The problem is to find the greatest possible density \(d_n (\mu)\) of n-dimensional Minkowski sphere arrangements of order μ. In this paper we give upper bounds on \(d_n (\mu)\) for \(\mu \le {1 \over n}\).
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Böröczky, K., Szabó, L. Minkowski Arrangements of Spheres. Monatsh. Math. 141, 11–19 (2004). https://doi.org/10.1007/s00605-002-0002-5
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DOI: https://doi.org/10.1007/s00605-002-0002-5