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On the section of a lattice-covering of balls. (English) Zbl 0543.52012
It is well-known that the density of any covering of the Euclidean plane by equal circles is at least 2$$\pi$$ /$$\sqrt{27}$$. This is not true for coverings by incongruent circles. The following theorem is proved: Let C be a lattice covering of Euclidean 3-space by unit spheres, and let p be a plane. Then the intersections of p with the spheres of C form a covering of p with density not less than $$2\pi /\sqrt{27}+\epsilon,$$ where $$\epsilon =0.0174.. .$$ This bound is attained by a uniquely determined lattice covering by spheres which is not that of minimal density. The plane p touches a sphere of this covering and is parallel to a face of a fundamental parallelepiped. The proof of this remarkable theorem is rather complicated and uses both geometrical and analytical arguments. If C is a non-lattice covering of space by equal spheres, the density of a plane section of C may be arbitrarily close to 1. The above theorem should be compared with a result by G. Fejes Tóth concerning the section of a packing of equal n-dimensional spheres with a sub-space [Studia Sci. Math. Hungar. 15, 487-489 (1980; Zbl 0518.52015)].
Reviewer: A.Florian

MSC:
 52C17 Packing and covering in $$n$$ dimensions (aspects of discrete geometry) 52A40 Inequalities and extremum problems involving convexity in convex geometry 11H31 Lattice packing and covering (number-theoretic aspects) 52C07 Lattices and convex bodies in $$n$$ dimensions (aspects of discrete geometry)
Zbl 0518.52015