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.

提供了最小外接球问题的数学基础背景，探讨了在d维欧几里德空间中确定最小半径包围给定有界集合的唯一球面。研究了与最小外接球问题相似或相关的多个问题，这些问题在科学技术的各个领域中得到了广泛应用。提出的理论框架基于几个包围和分区定理，提供了环周半径、内接半径、直径和集合宽度之间的界限和关系。这些包围和分区定理被视为该领域的基石，对其他空间和非欧几里德几何的发展和推广产生了重要影响。

最小外接球及相关问题的数学基础研究