  In 1971, McMullen and Walkup posed the following conjecture, which is called
the generalized lower bound conjecture: If $P$ is a simplicial $d$-polytope
then its $h$-vector $(h_0,h_1,...,h_d)$ satisfies $h_0 \leq h_1 \leq ... \leq
h_{\lfloor \frac d 2 \rfloor}$. Moreover, if $h_{r-1}=h_r$ for some $r \leq
\frac d 2$ then $P$ can be triangulated without introducing simplices of
dimension $\leq d-r$.
  The first part of the conjecture was solved by Stanley in 1980 using the hard
Lefschetz theorem for projective toric varieties. In this paper, we give a
proof of the remaining part of the conjecture. In addition, we generalize this
property to a certain class of simplicial spheres, namely those admitting the
weak Lefschetz property.
