  Projective Reed-Muller codes correspond to subcodes of the Reed-Muller code
in which the polynomials being evaluated to yield codewords, are restricted to
be homogeneous. The Generalized Hamming Weights (GHW) of a code ${\cal C}$,
identify for each dimension $\nu$, the smallest size of the support of a
subcode of ${\cal C}$ of dimension $\nu$. The GHW of a code are of interest in
assessing the vulnerability of a code in a wiretap channel setting. It is also
of use in bounding the state complexity of the trellis representation of the
code.
  In prior work by the same authors, a code-shortening algorithm was employed
to derive upper bounds on the GHW of binary projective, Reed-Muller (PRM)
codes. In the present paper, we derive a matching lower bound by adapting the
proof techniques used originally for Reed-Muller (RM) codes by Wei. This
results in a characterization of the GHW hierarchy of binary PRM codes.
