On the location of geometrical medians of triangles

The geometrical median is a natural spatial generalization of the statistical median of a one-dimensional sample. Thus the problem of computing the median of a finite set of points (a sample) on a straight line presents no difficulties, but unexpected difficulties arise in moving to the plane or to higher dimensional spaces, where the natural linear order of points is absent. While the mean of a multidimensional sample, as on a straight line, is calculated by taking the arithmetic mean, no such analytical formula is available for the geometric median. Moreover, such formulas are absent when we deal with geometrical medians of continuous objects located on a plane or in space. This raises the natural question of analytical estimates of the locations of geometric medians. This paper presents the solutions for two such simplest problems. Namely, the solution of the problem on estimating the location of the geometric median of the perimeter of a triangle and the solution of a similar problem on the geometric median of a triangular area. For both problems, we obtain exact estimates of the affine type.