欧氏球的广义Sylvester问题和广义Fermat-Torricelli问题解的构造
Solutions Constructions of a Generalized Sylvester Problem and a Generalized Fermat-Torricelli Problem for Euclidean Balls
摘要：
经典的阿波罗尼乌斯问题是在一个平面上构造与三个给定圆相切的圆。这个问题是由阿波罗尼乌斯Perga在他的工作“切线”。西尔维斯特问题（英语：Sylvester problem）是由英国数学家J. J.西尔维斯特（英语：J. J. Sylvester）提出的，要求在平面上包围有限个点的最小圆。本文研究了Sylvester问题的以下推广形式及其与Apollonius问题的联系：给定$\Bbb R^n$中两个有限的欧几里得球集合，找出包围第一个集合中所有球并与第二个集合中所有球相交的最小欧几里得球.我们还研究了一个广义版本的费马-托里切利问题如下所述：给定两个有限的集合组成的三个欧几里德球在$\Bbb R^n$，找到一个点，最小化的总和的最远距离的球在第一个集合和最短距离的球在第二个集合。
英文摘要：
The classical Apollonius' problem is to construct circles that are tangent to three given circles in a plane. This problem was posed by Apollonius of Perga in his work "Tangencies". The Sylvester problem, which was introduced by the English mathematician J.J. Sylvester, asks for the smallest circle that encloses a finite collection of points in the plane. In this paper, we study the following generalized version of the Sylvester problem and its connection to the problem of Apollonius: given two finite collections of Euclidean balls in $\Bbb R^n$, find the smallest Euclidean ball that encloses all of the balls in the first collection and intersects all of the balls in the second collection. We also study a generalized version of the Fermat-Torricelli problem stated as follows: given two finite collections composed of three Euclidean balls in $\Bbb R^n$, find a point that minimizes the sum of the farthest distances to the balls in the first collection and shortest distances to the balls in the second collection.
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