For a given point-arrangement construct the convex hull or the tangential polyhedron. If the convex hull has non-triangle-facets make it a triangular polyhedron by drawing diagonals. Separate factes of the tangential polyhedron also into triangles by connecting the vertices of the facet with the point of contact.
Thus we may assume that the polyhedron consists the triangles Δ1, ... , Δp. Let be
three functions onto R.
m: Rp -> R ("measuring function") D: S2 x S2 x S2 -> R ("triangle-function") e: S2 x S2 -> R ("edge-function "),
Then the evaluation of the point-arrangement using a polyhedral construction is given by:F = F(x1,..xn) := m( D(Δ1),..., D(Δp) )
If the evaluation of the point-arrangement uses the distancies of the points, the evaluation is given by:G = G(x1,..xn) := m( e(p1,p1), e(p1,p2),..., e(pn,pn) )
We always may assume the following "natural conditions " to these functions: