The Fermat-Weber Problem and Inner-Product Spaces

Facility location in normed linear spaces

“…The following papers studied the Fermat-Torricelli problem in normed linear spaces: Alexandrescu [1], Durier and Michelot [9,11] and Vesely [27]. Durier and Michelot in [9,11] studied geometric properties of the Fermat-Weber point in spaces with scalar product, and Vesely studied generalized Chebyschev centers in [27].…”

“…Durier and Michelot in [9,11] studied geometric properties of the Fermat-Weber point in spaces with scalar product, and Vesely studied generalized Chebyschev centers in [27]. Alexandrescu [1] studied the Fermat point problem for a system of n distinct points in Hilbert spaces.…”

Location of the Fermat-Torricelli medians of three points

“…Some partial answers on the subject of this paper have been obtained by Durier [8], who proved that a characteristic of real inner product spaces of dimension ≥ 3 is the hypothesis…”

On Generalizations of Conics and on a Generalization of the Fermat- Torricelli Problem

“…From this treatment we get another generalization of conics in the plane. The results in this article are also interesting in connection with the problem of determining the Fermat-Torricelli points [1], [61, [121. We construct examples of configurations of focal points in certain norms where the Fermat-Torricelli set contains points that do not lie in the convex hull of the focal points [7]. This definition holds for arbitrary dimensions.…”

“…Drawing routines from computer-algebra systems such as Mathematica and Maple can be used to obtain pictures of generalized conics. With Mathematica it is possible to draw pictures for the 1-dimensional case, e.g., in the 2-norm, using the following commands; see Figures 6,7: (* Fig 6* In addition, Maple can draw graphs of implicitly defined functions, so we can also get results in the 2-dimensional case; see Figure 5 (right). 4.…”