Density of an $(r,R)$-system

In this paper we give a complete geometrical theory for the study of the exact lower bound of the density of $n$-dimensional lattices. For arbitrary $(r,R)$-systems we prove an analog of well known theorems due to Rogers from the theory of packings, and also from this same theory, an analog of a theorem due to Coxeter, Few, and Rogers. Several special examples are treated.