On the existence of completely saturated packings and completely reduced covering

الأصل · EN

A packing by a body K is collection of congruent copies of K (in either Euclidean or hyperbolic space) so that no two copies intersect nontrivially in their interiors. A covering by K is a collection of congruent copies of K such that for every point p in the space there is copy in the collection containing p. A completely saturated packing is one in which it is not possible to replace a finite number of bodies of the packing with a larger number and still remain a packing. A completely reduced covering is one in which it is not possible to replace a finite number of bodies of the covering with a smaller number and still remain a covering. It was conjectured by G. Fejes Toth, G. Kuperberg, and W. Kuperberg that completely saturated packings and commpletely reduced coverings exist for every body K in either n-dimensional Euclidean or n-dimensional hyperbolic space. We prove this conjecture.