  We consider the construction of a polyhedral Delaunay partition as a limit of
the sequence of power diagrams (radical partitions). The dual Voronoi diagram
is obtained as a limit of the sequence of weighted Delaunay partitions. The
problem is reduced to the construction of two dual convex polyhedra, inscribed
and superscribed around a circular paraboloid, as a limit of the sequence of
pairs of general dual convex polyhedra. The sequence of primal polyhedra should
converge to the superscribed polyhedron and the sequence of the dual polyhedra
converges to the inscribed polyhedron.
  We are interested in the case when the vertices of primal polyhedra can move
or merge together, i.e., no new faces are allowed for dual polyhedra. These
rules define the transformation of the set of initial spheres into the set of
Delaunay spheres using radius variation and sphere movement and elimination.
Existence theorems are still unavailable but we suggest a functional measuring
the deviation of the convex polyhedron from the one inscribed into the
paraboloid. It is the discrete Dirichlet functional for the power function
which is a linear interpolant of the vertical distance of the dual vertices
from the paraboloid. The functional's absolute minimizer is attained on the
constant power field, meaning that the inscribed polyhedron can be obtained by
a simple translation. This formulation of the functional for the dual surface
is not quadratic since the unknowns are the vertices of the primal polyhedron,
hence, the transformation of the set of spheres into Delaunay spheres is not
unique.
  We concentrate on the experimental confirmation of the approach viability and
put aside mesh quality problems. The zero value of the gradient of the proposed
functional defines a manifold describing the evolution of Delaunay spheres.
Hence, Delaunay-Voronoi meshes can be optimized using the manifold as a
constraint.
