A new separable property of the joint numerical range of quadratic functions and its applications to the Smallest Enclosing Ball Problem

TL;DR

This paper investigates the separable property of the joint numerical range of quadratic functions and applies it to improve solutions for the Smallest Enclosing Ball problem, especially in cases where the joint numerical range is non-convex.

Contribution

It introduces a new set $G(R^n)^ullet$ that exhibits convexity even when the original joint numerical range is non-convex, enabling better problem-solving strategies for the SEB problem.

Findings

Convexity of $G(R^n)$ depends on the rank condition of the points.

The set $G(R^n)^ullet$ is convex when $m=n$, even if $G(R^n)$ is not.

Separable property of $G(R^n)^ullet$ implies separability for $G(R^n)$, aiding in solving the SEB problem.

Abstract

We explore separable property of the joint numerical range $G(\Bbb R^n)$ of a special class of quadratic functions and apply it to solving the smallest enclosing ball (SEB) problem which asks to find a ball $B(a,r)$ in $\Bbb R^n$ with smallest radius $r$ such that $B(a,r)$ contains the intersection $\cap_{i=1}^mB(a_i,r_i)$ of $m$ given balls $B(a_i,r_i).$ We show that $G(\Bbb R^n)$ is convex if and only if ${\rm rank}\{a_1-a, a_2-a, \ldots, a_m-a\}\le n-1.$ Otherwise, ${\rm rank}\{a_1-a, a_2-a, \ldots, a_m-a\}=n$ and $G(\Bbb R^n)$ is not convex. In this case we propose a new set $G(\Bbb R^n)^\bullet$ which allows to show that if $m=n$ then $G(\Bbb R^n)^\bullet$ is convex even $G(\Bbb R^n)$ is not. Importantly, the separable property of $G(\Bbb R^n)^\bullet$ then implies the separable property for $G(\Bbb R^n).$ As a result, a new progress on solving the SEB problem is obtained.