  A remarkable and important property of face numbers of simplicial polytopes
is the generalized lower bound inequality, which says that the $h$-numbers of
any simplicial polytope are unimodal. Recently, for balanced simplicial
$d$-polytopes, that is simplicial $d$-polytopes whose underlying graphs are
$d$-colorable, Klee and Novik proposed a balanced analogue of this inequality,
that is stronger than just unimodality. The aim of this article is to prove
this conjecture of Klee and Novik. For this, we also show a Lefschetz property
for rank-selected subcomplexes of balanced simplicial polytopes and thereby
obtain new inequalities for their $h$-numbers.
