There is a system of n points in 3-D space. For the "overall average" -characteristic of this points, two point estimates of the location are usually used: the geometric median and the geometric center (or center of mass, centroid). In the first case, it is a point that minimizes the function of the sum of the distances to a set of n points -Center of Minimum Distance (for three points, this is the Fermat point), and the sum of the squared distances -in the second case.
I propose a more extended interpretation of the definitions of these two point estimates of the location of a discrete set of n points. These estimates are made, e.g. not in ℝ³, but on a restricted domain. That is, you need to find points in this domain that extremize the function of the sum of the distances (or squared distances) to all points of set.
In applets, a limited number of particles from ℝ³ are considered, and a circle and a sphere are considered as such a domain, i.e. point estimates are sought on a circle / sphere. The search for point estimates on a circle allows you to understand their meaning more clearly . Thus, the search for point estimates is reduced to finding the critical points of the distance sum function f (x, y, z) subject to a constraining equation g(x,y)=x²+y -R² in the case of estimation on a circle or g(x,y,z)=x²+y²+z²-R²-in the case of estimation on the sphere. The problem is solved using Lagrange multipliers. There is a system of equations: ∇f(x,y,z)= λ∇g(x,y,z). A local optimum occurs when ∇f(x,y,z) and ∇g(x,y,z) are parallel, and so ∇f is some multiple of ∇g.
Algorithms are proposed for finding points corresponding to extremes: minima, maxima, and, in the case of the sphere, also saddle points of the sum of distances function.
From: List of My Public Books on GeoGebra Topics: Constructing polyhedra -[url]https://www.geogebra.org/m/eabstecp[/url]
