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Compact packings in the Euclidean space. (English) Zbl 0619.52010

A packing of convex bodies in \({\mathbb{E}}^ n\) is said to be compact if any continuous curve which connects an arbitrary body of the packing with a point sufficiently distant from the body, intersects at least one body of the packing touching the body under consideration. L. Fejes Tóth (oral communication) proved that the lower density of a compact packing of homothetic centrally symmetric convex discs with ratios of diameters bounded is at least 3/4. If the central symmetry is dropped then the lower density is at least 1/2 as shown by A. Bezdek, the author and K. Böröczky [Stud. Sci. Math. Hung., to appear]. The author shows that the density of a compact lattice packing of a centrally symmetric convex body in \({\mathbb{E}}^ n\) is greater than \(2^{1/(n-1)}/(2^{1/(n-1)}+1).\)
Reviewer: P.Gruber

MSC:

52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry)
52A40 Inequalities and extremum problems involving convexity in convex geometry
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