  In this paper, we determine the topology of the spaces of convex polyhedra
inscribed in the unit $2$-sphere and the spaces of strictly Delaunay geodesic
triangulations of the unit $2$-sphere. These spaces can be regarded as
discretized groups of diffeomorphisms of the unit $2$-sphere. Hence, it is
natural to conjecture that these spaces have the same homotopy types as those
of their smooth counterparts. The main result of this paper confirms this
conjecture for the unit $2$-sphere. It follows from an observation on the
variational principles on triangulated surfaces developed by I. Rivin.
  On the contrary, the similar conjecture does not hold in the cases of flat
tori and convex polygons. We will construct simple examples of flat tori and
convex polygons such that the corresponding spaces of Delaunay geodesic
triangulations are not connected.
