  An attempt to construct the ``ground state'' vacuum initial data for the
gravitational field surrounding two black holes is presented. The ground state
is defined as the gravitational initial data minimizing the ADM mass within the
class of data for which the masses of the holes and their distance are fixed.
To parameterize different geometric arrangements of the two holes (and,
therefore, their distance) we use an appropriately chosen scale factor. A
method for analyzing the variations of the ADM mass and the masses (areas) of
the horizons in terms of gravitational degrees of freedom is proposed. The
Misner initial data are analyzed in this context: it is shown that they do not
minimize the ADM mass.
