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.

翻译：本文提供了最小包围球问题数学基础的理论背景。该问题涉及在d维欧几里得空间中，确定包围给定有界集合的唯一最小半径球面。由于最小包围球及其相似或相关问题在科学技术各领域具有广泛应用，相关研究获得了显著推动。提出的理论框架基于若干包围（覆盖）与分割（聚类）定理，为集合的外接半径、内切半径、直径与宽度提供了界限及相互关系。这些包围与分割定理被视为该领域的奠基性成果，对后续向其他空间与非欧几何的拓展与泛化产生了深远影响。