  Theoretical background is provided towards the mathematical foundation of the
minimum enclosing ball problem. This problem concerns the determination of the
unique spherical surface of smallest radius enclosing a given bounded set in
the d-dimensional Euclidean space. The study of several problems that are
similar or related to the minimum enclosing ball problem has received a
considerable impetus from the large amount of applications of these problems in
various fields of science and technology. The proposed theoretical framework is
based on several enclosing (covering) and partitioning (clustering) theorems
and provides among others bounds and relations between the circumradius,
inradius, diameter and width of a set. These enclosing and partitioning
theorems are considered as cornerstones in the field that strongly influencing
developments and generalizations to other spaces and non-Euclidean geometries.
