Solutions Constructions of a Generalized Sylvester Problem and a Generalized Fermat-Torricelli Problem for Euclidean Balls

TL;DR

This paper introduces new solutions for generalized Sylvester and Fermat-Torricelli problems involving Euclidean balls, extending classical geometric problems to more complex configurations and providing constructive methods.

Contribution

It develops novel solution constructions for generalized Sylvester and Fermat-Torricelli problems involving Euclidean balls, linking classical geometry with modern optimization.

Findings

Derived explicit solution methods for the generalized Sylvester problem.

Established connections between Sylvester and Apollonius problems.

Proposed solutions for the generalized Fermat-Torricelli problem with Euclidean balls.

Abstract

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.