二次函数联合数值范围的一个新可分性质及其在最小包围球问题中的应用
A new separable property of the joint numerical range of quadratic functions and its applications to the Smallest Enclosing Ball Problem
AI总结：
本文研究一类二次函数联合数值范围的可分性，提出新集合$G(\Bbb R^n)^\bullet$，在秩条件为$n$时仍保持凸性，从而推动最小包围球问题的求解。
摘要：
我们探讨了一类特殊二次函数的联合数值范围$G(\Bbb R^n)$的可分性质，并将其应用于求解最小包围球（SEB）问题，该问题要求找到$\Bbb R^n$中半径$r$最小的球$B(a,r)$，使得$B(a,r)$包含$m$个给定球$B(a_i,r_i)$的交集$\cap_{i=1}^mB(a_i,r_i)$。我们证明$G(\Bbb R^n)$是凸的当且仅当${\rm rank}\{a_1-a, a_2-a, \ldots, a_m-a\}\le n-1$。否则，${\rm rank}\{a_1-a, a_2-a, \ldots, a_m-a\}=n$且$G(\Bbb R^n)$不是凸的。在这种情况下，我们提出一个新的集合$G(\Bbb R^n)^\bullet$，它能够表明：若$m=n$，则即使$G(\Bbb R^n)$不凸，$G(\Bbb R^n)^\bullet$也是凸的。重要的是，$G(\Bbb R^n)^\bullet$的可分性质进而蕴含$G(\Bbb R^n)$的可分性质。由此，我们在求解SEB问题上取得了新的进展。
英文摘要：
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.
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