  In this work, a lower bound for the ground state energy of the
Falicov-Kimball model for intermediate densities is derived. The explicit
derivation is important in the proof of the conjecture of segregation of the
two kinds of fermions in the Falicov-Kimball model, for sufficiently large
interactions. This bound is given by a bulk term, plus a term proportional to
the boundary of the region devoid of classical particles. A detailed proof is
presented for density n=1/2, where the coefficient 10^(-13) is obtained for the
boundary term, in two dimensions. With suitable modifications the method can
also be used to obtain a coefficient for all densities.
