Towards the mathematical foundation of the minimum enclosing ball and related problems
This foundational work addresses a core geometric problem with broad applications in science and technology, but it is incremental as it builds on existing theorems.
The paper tackles the mathematical foundation of the minimum enclosing ball problem, providing a theoretical framework based on enclosing and partitioning theorems that yield bounds and relations between geometric measures like circumradius and diameter.
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.