The weighted Fermat triangle problem
Let there be given a triangle \(p_1p_2p_3\) and three positive numbers \(\lambda_1, \lambda_2, \lambda_3\), called the weights of \(p_1, p_2,\) and \(p_3\), respectively. The weighted Fermat-Torricelli problem (in location science also called the Steiner-Weber problem) asks for a point \(p\) minimizing the sum \[ \sum_{i=1}^3\lambda_i\| p_i-p\| . \] For \(m\) non-collinear points \(p_1, \ldots, p_m\in\mathbb{R}^n\), \(n\geq 2\), with corresponding positive weights \(\lambda_1, \ldots\lambda_m\) there exists an unique point \(p\in \mathbb{R}^n\) that minimizes the sum \[ \sum_{i=1}^m\lambda_i\| p_i-p\| ; \] see, e.g., Theorem 18.37 in [\textit{V. Boltyanski, H. Martini} and \textit{V. Soltan}, Geometric methods and optimization problems, Combinatorial Optimization. 4, Dordrecht: Kluwer Academic Publishers (1999; Zbl 0933.90002)]. More about the location of the point \(p\) as well as about different aspects and history of this famous problem can be found in Chapter 2 of [loc. cit.]. In case \(\lambda_1=\dots=\lambda_m=1\) the Fermat-Torricelli problem is even generalized for an arbitrary normed linear space [\textit{H. Martini, K. J. Swanepoel} and \textit{G. Weiss}, J. Optimization Theory Appl. 115, No. 2, 283--314 (2002; Zbl 1047.90032)]. In the paper under review a solution is given for the usual Euclidean norm in the case \(n=2\) and \( m=3\).
scientific article; zbMATH DE number 5313278
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