2012/10/11 by Nguyen Mau Nam, Nam, Nguyen Mau, Nguyen Hoang +3
Engineering · Mathematics · Physics and Astronomy · #Advanced Mathematical Theories and Applications #Advanced Theoretical and Applied Studies in Material Sciences and Geometry #FOS: Mathematics #Mathematics and Applications #Optimization and Control (math.OC) #math.OC
paper · pdf · doi:10.48550/arxiv.1210.3142
arxiv created 2012/10/11 · openalex publication_date 2012/10/11 · arxiv updated 2012/10/12 · openalex created_date 2022/08/31 · openalex updated_date 2026/07/28
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 Rn, 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 Rn, 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.