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A geometric generalization of the Vecten-Fasbender duality. (English) Zbl 0906.90106
The Fermat-Toricelli problem is to find for three given points $a_1$, $a_2$ and $a_3$ another point which minimizes the sum of the distances to the three given points. Vecten (1811/12) and Fasbender (1846) observed that the perpendiculars to the lines connecting the Fermat-Toricelli point to the three given points $a_1$, $a_2$ and $a_3$ are the sides of the maximal equilateral triangle circumscribing these three points. This duality can be seen as the first example of dualizing a problem in the spirit of nonlinear programming. The author generalizes this duality to the case where the given points may have weights assigned. He proves a geometric characterization of this generalized Vecten-Fasbender duality. Moreover, he gives an overview of the historical development of this problem, including the analytical approach of Kuhn (1967, 1974).
90B85Continuous location