  Several stars orbit around a black hole candidate of mass $3.7\times 10^6$
M$_{\odot}$, in the region of the Galactic Center (GC). Looking for General
Relativistic (GR) periastron shifts is limited by the existence of a stellar
cluster around the black hole that would modify the orbits due to classical
effects that might mask the GR effect. Only if one knows the cluster parameters
(its mass and core radius) it is possible to unequivocally deduce the GR
effects expected and then test them. In this paper it is shown that the
observation of the proper motion of Sgr A$^*$, $v_{Sgr A^*} = (0.4\pm 0.9)$ km
s$^{-1}$ (\citealt{reid2004}), could help us to constrain the cluster
parameters significantly and that future measurements of the periastron shifts
for at least three stars may adequately determine the cluster parameters and
the mass of the black hole.
