[size=85][color=#333333] Let P[/color]i[color=#333333]=(x[/color]i[color=#333333],y[/color]i[color=#333333],z[/color]i[color=#333333]) n moving points in ℝ³ (lP:={[/color]P1,P2,...,Pn[color=#333333]}). I want to find  the points P=(x,y,z) [i][u]on the surface of the sphere[/u][/i] -S ([/color][i]radius R[/i][color=#333333]) that are critical (relative [/color][color=#0000ff]min[/color][color=#333333]/[/color][color=#ff0000]max[/color][color=#333333] or [/color][color=#6aa84f]saddle [/color][color=#333333]points at (x,y,z)) of a function [/color][color=#333333][color=#1e84cc]f(x,y,z)[/color][i]:=-[/i][/color][color=#1e84cc][i]sum of the distances[/i][/color][color=#333333] from P to the all points from lP. [/color][color=#333333]Critical points can be found using [/color][i]Lagrange multipliers[/i][i]as [/i]finding the Extreme values of the function [color=#1e84cc]f(x,y,z) [/color][i]subject to [/i][i]a constraining equation [/i][color=#333333]g(x,y,z):=x[/color][sup]2[/sup][color=#333333]+y[/color][sup]2[/sup][color=#333333]+z[/color][sup]2[/sup][color=#333333]-R[/color][sup]2[/sup][color=#333333]=0. There is a system of equations: ∇[/color][i][color=#1e84cc]f(x,y,z)[/color][/i][color=#333333]= λ∇g(x,y,z). A local optimum occurs when ∇[/color][i][color=#1e84cc]f(x,y,z)[/color][/i][color=#333333] and ∇g(x,y,z) are parallel[/color], and so ∇[i][color=#1e84cc]f[/color][/i] [color=#000000]is some multiple of ∇[/color][i][color=#000000][i]g[/i][/color][/i][color=#000000]. [br] This applet is used to study the distribution of geometric medians on a sphere of radius R, „induces“ by the discrete sample of 3 movable points in the 3-D space. [br]Description is in [url=https://www.geogebra.org/m/y8dnkeuu]https://www.geogebra.org/m/y8dnkeuu[/url]. [/color][/size][size=85]The type of critical points is specified using the hessian matrix in the [url=https://www.geogebra.org/m/bx5gs7qt]applet[/url].[/size]