Constructions of Solutions to Generalized Sylvester and Fermat–Torricelli Problems for 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 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 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.

A proximal method for solving nonlinear minmax location problems with perturbed minimal time functions via conjugate duality

“…where a i > 0 and p i ∈ R 2 , i = 1, 2, 3, then this problem is also known as the classical Apollonius problem (see [4,13,18]).…”

Minsum Location Extended to Gauges and to Convex Sets

“…Can one characterize other geometric properties of the solution set? A complete characterization of the Fermat-Torricelli locus of three Euclidean balls with distinct centers in Euclidean space can be found in [26,Sect. 4.1].…”