The Smallest Enclosing Ball Problem and the Smallest Intersecting Ball Problem: Existence and Uniqueness of Solutions

TL;DR

This paper investigates the existence and uniqueness of solutions for the smallest enclosing and intersecting ball problems in normed spaces, generalizing classical circle problems and establishing foundational conditions for optimal solutions.

Contribution

It provides new sufficient conditions for the existence and uniqueness of solutions to these generalized ball problems in normed spaces.

Findings

Established conditions for solution existence

Proved uniqueness under certain conditions

Generalized classical circle problem to normed spaces

Abstract

In this paper we study the following problems: given a finite number of nonempty closed subsets of a normed space, find a ball with the smallest radius that encloses all of the sets, and find a ball with the smallest radius that intersects all of the sets. These problems can be viewed as generalized versions of the smallest enclosing circle problem introduced in the 19th century by Sylvester which asks for the circle of smallest radius enclosing a given set of finite points in the plane. We will focus on the sufficient conditions for the existence and uniqueness of an optimal solution for each problem, while the study of optimality conditions and numerical implementation will be addressed in our next projects.