Geometric median in the plane (Q2341524)

Given \(\{T_i,1\leq i\leq m\}\) a set of points in the plane with corresponding weights \(w_i>0\), the weighted geometric median of these points is the point \(T\) minimizing the sum of the weighted Euclidean distances from \(T\) to the previous points. In this paper, the authors study the problem of determining the geometric median of three points; i.e., the weighted geometric median with all weights equal to 1. In particular they relate this point to the Torricelli point of the triangle formed by the three points (provided they are not collinear). This is done by purely geometric techniques. For higher dimensional settings, the authors translate the geometric problem to that of minimizing a functional \(F:\mathbb{R}^2\rightarrow\mathbb{R}\). To do so they apply Weiszfeld's algorithm. Finally, some examples are given.