Compact packings in the Euclidean space
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 \(2^{1/(n-1)}/(2^{1/(n-1)}+1).\)
scientific article; zbMATH DE number 4005053