Related papers: The Fermat-Torricelli problem in normed spaces
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,…
We investigate the Fermat-Torricelli problem in d-dimensional real normed spaces or Minkowski spaces, mainly for d=2. Our approach is to study the Fermat-Torricelli locus in a geometric way. We present many new results, as well as give an…
One of the oldest and richest problems from continuous location science is the famous Fermat-Torricelli problem, asking for the unique point in Euclidean space that has minimal distance sum to n given (non-collinear) points. Many natural…
In this paper we develop new applications of variational analysis and generalized differentiation to the following optimization problem and its specifications: given n closed subsets of a Banach space, find such a point for which the sum of…
Let $P_1,P_2,P_3$ be three given points in $\mathbf{R}^2$, and $P$ be an arbitrary point in $\mathbf{R}^2$. The classical Fermat's problem to Torricelli asks for the location of the point $P$ such that $|PP_1|+|PP_2|+|PP_3|$ is a minimum.…
We prove the following fundamental property for the Fermat-Torricelli point for four non-collinear and non-coplanar points forming a tetrahedron in $\mathbb{R}^{3},$ which states that: The three bisecting lines having as a common vertex the…
In the early 17th century, Pierre de Fermat proposed the following problem: given three points in the plane, find a point such that the sum of its Euclidean distances to the three given points is minimal. This problem was solved by…
In this paper, we will establish a general method of studying finite-dimensional normed spaces, and apply this method to classifying $3$-dimensional and $4$-dimensional normed spaces over a non-spherically complete field. For this purpose,…
The weighted Fermat-Torricelli problem for four non-collinear and non-coplanar points in the three dimensional Euclidean Space states that: Given four non-collinear and non-coplanar points A1, A2, A3, A4 and a positive real number (weight)…
We obtain two analytic solutions for the weighted Fermat-Torricelli problem in the Euclidean Plane which states that: Given three points in the Euclidean plane and a positive real number (weight) which correspond to each point, find the…
We obtain an important generalization of the inverse weighted Fermat-Torricelli problem for tetrahedra in R^3 by assigning at the corresponding weighted Fermat-Torricelli point a remaining positive number (residual weight). As a…
In this paper, we introduce and study the following problem and its further generalizations: given two finite collections of sets in a normed space, find a ball whose center lies in a given constraint set with the smallest radius that…
We solve the problem of best approximation by partial isometries of given rank to an arbitrary rectangular matrix, when the distance is measured in any unitarily invariant norm. In the case where the norm is strictly convex, we parametrize…
We discuss the computational complexity of special cases of the 3-dimensional (axial) assignment problem where the elements are points in a Cartesian space and where the cost coefficients are the perimeters of the corresponding triangles…
We obtain an analytical solution for the weighted Fermat-Torricelli problem for an equilateral geodesic triangle A_1A_2A_3 which is composed by three equal geodesic arcs (sides) of length Pi/2 for given three positive unequal weights that…
The problem of replacing an integral norm with respect to a given probability measure by the corresponding integral norm with respect to a discrete measure is discussed in the paper. The above problem is studied for elements of finite…
In this paper, we begin by introducing a well-known geometry concept: the Fermat point in a triangle. Then, we generalize the problem and propose an iterative algorithm based on gradient descent to the weighted form in Lp space. We also…
This paper explores the solution of Fredholm-like equations with infinite dimensional solution spaces. We set out to find a method for determining a particular solution to a Fredholm-like equation subject to a given constraint. The…
We study a generalization of the weighted Fermat-Torricelli problem in the plane, which is derived by replacing vertices of a convex polygon by 'small' closed convex curves with weights being positive real numbers on the curves, we also…
In this paper, we prove several fixed point theorems on both of normal partially ordered Banach spaces and regular partially ordered Banach spaces by using the normality, regularity, full regularity, and chain -complete property. Then, by…