87 research outputs found
The Fermat-Torricelli problem in the case of three-point sets in normed planes
In the paper the Fermat-Torricelli problem is considered. The problem asks a
point minimizing the sum of distances to arbitrarily given points in
d-dimensional real normed spaces. Various generalizations of this problem are
outlined, current methods of solving and some recent results in this area are
presented. The aim of the article is to find an answer to the following
question: in what norms on the plane is the solution of the Fermat-Torricelli
problem unique for any three points. The uniqueness criterion is formulated and
proved in the work, in addition, the application of the criterion on the norms
set by regular polygons, the so-called lambda planes, is shown.Comment: 13 pages, 9 figure
Isometric embedding of a weighted Fermat-Frechet multitree for isoperimetric deformations of the boundary of a simplex to a Frechet multisimplex in the -Space
In this paper, we study the weighted Fermat-Frechet problem for a tuple of positive real numbers determining -simplexes in the
dimensional -Space (-dimensional Euclidean space if
the -dimensional open hemisphere of radius
() if and the Lobachevsky space
of constant curvature if ). The (weighted)
Fermat-Frechet problem is a new generalization of the (weighted) Fermat problem
for -simplexes. We control the number of solutions (weighted Fermat trees)
with respect to the weighted Fermat-Frechet problem that we call a weighted
Fermat-Frechet multitree, by using some conditions for the edge lengths
discovered by Dekster-Wilker. In order to construct an isometric immersion of a
weighted Fermat-Frechet multitree in the - Space, we use the isometric
immersion of Godel-Schoenberg for -simplexes in the -sphere and the
isometric immersion of Gromov (up to an additive constant) for weighted Fermat
(Steiner) trees in the -hyperbolic space . Finally, we
create a new variational method, which differs from Schafli's, Luo's and
Milnor's techniques to differentiate the length of a geodesic arc with respect
to a variable geodesic arc, in the 3-Space. By applying this method, we
eliminate one variable geodesic arc from a system of equations, which give the
weighted Fermat-Frechet solution for a sextuple of edge lengths determining
(Frechet) tetrahedra.Comment: 47 pages, 1 figur
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