arXiv 1111.12802011-11math.OC

最小闭球问题与最小交球问题：解的存在唯一性

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

Boris S. Mordukhovich, Nguyen Mau Nam, Cristina Villalobos

摘要：

本文研究了以下问题：给定赋范空间的有限个非空闭子集，求一个包围所有集合的半径最小的球，以及求一个与所有集合相交的半径最小的球。这些问题可以看作是广义版本的最小封闭圆问题介绍了在19世纪西尔维斯特要求圆的最小半径封闭一组给定的有限点在平面上。我们将专注于每个问题的最优解的存在性和唯一性的充分条件，而最优性条件和数值实现的研究将在我们的下一个项目。

英文摘要：

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.

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