ИСТИНА

The talk discusses a generalization of the classical Fermat- -Torricelli problem to normed spaces of arbitrary finite dimension. The problem of finding a point that minimizes the sum of distances from it to a given set of points in a metric space was first mentioned in the 17th century. In 1643 Fermat posed a problem for three points on the Euclidean plane, and in the same century Torricelli proposed a solution to this problem. Since then, various generalizations of this problem have been considered. The problem was formulated for an arbitrary number of points, the dimension of the space, as well as the norm given in this space. The simplicity of the formulation allows us to consider the problem even in an arbitrary metric space. The work describes the application of a geometric approach to the problem and presents some new results that are obtained in the framework of real finite-dimensional normed spaces, called Minkowski spaces. The aim of this talk is to present necessary and sufficient conditions for the uniqueness of the solution of the Fermat- -Torricelli problem for any n points in a fixed space, and more precise conditions for normed planes and three- dimensional spaces. In addition, examples of the application of the criterion in the norms given by regular polygons and regular polyhedra are given.