## Formalisation of the Evaluations

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
 m: Rp -> R ("measuring function") D: S2 x S2 x S2 -> R ("triangle-function") e: S2 x S2 -> R ("edge-function "),
three functions onto R.

Then the evaluation of the point-arrangement using a polyhedral construction is given by:

F = F(x1,..xn) := m( D1),..., Dp) )

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:

• m, D and e are continuous, in most cases they are also differentable.
• m > 0. Often m is a norm, e.g. lp
• The value of m does not depend on the order of coordinates; i.e.:
m( ..xi-1,xi, xi+1,..., xj-1,xj, xj+1,.. ) = m( ..xi-1,xj, xi+1,..., xj-1,xi, xj+1,.. ),
• D( p1,p2,p3 ) > 0 for each positive orientated triangle p1,p2,p3.
• In most cases D fullfills a "kind of monotony":
Among all triangles with given base AB the isoscele has extremal value (valued by D).
• e( p1,p2 ) > 0, equality if and only if p1 = p2
Lienhard Wimmer ( Impressum )
2018-05-19