  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.
