The Fermat–Torricelli Problem, Part I: A Discrete Gradient-Method Approach

Yaakov S. Kupitz (), Horst Martini () and Margarita Spirova ()
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Yaakov S. Kupitz: The Hebrew University of Jerusalem
Horst Martini: University of Technology
Margarita Spirova: University of Technology

Journal of Optimization Theory and Applications, 2013, vol. 158, issue 2, No 1, 305-327

Abstract: Abstract We give a discrete geometric (differential-free) proof of the theorem underlying the solution of the well known Fermat–Torricelli problem, referring to the unique point having minimal distance sum to a given finite set of non-collinear points in d-dimensional space. Further on, we extend this problem to the case that one of the given points is replaced by an affine flat, and we give also a partial result for the case where all given points are replaced by affine flats (of various dimensions), with illustrative applications of these theorems.

Keywords: Affine flats; Cauchy–Schwarz inequality; Discrete gradient method; Fasbender duality; Fermat–Torricelli problem; Location science; Multifocal ellipses; Steiner minimal trees; Steiner–Weber problem; Varignon frame (search for similar items in EconPapers)
Date: 2013
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Citations: View citations in EconPapers (5)

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DOI: 10.1007/s10957-013-0266-z

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