We prove that the order complex of a geometric lattice has a convex
ear decomposition. As a consequence, if
(L) is the order
complex of a rank (r + 1) geometric lattice L, then for all i
r/2 the h-vector of
(L) satisfies h
i-1 h
i
and h
i h
r-i.
We also obtain several inequalities for the flag h-vector of
(L) by analyzing the weak Bruhat order of the symmetric
group. As an application, we obtain a zonotopal
cd-analogue
of the Dowling–Wilson characterization of geometric lattices which
minimize Whitney numbers of the second kind. In addition, we are
able to give a combinatorial flag h-vector proof of h
i-1
h
i when i
(2/7)(r + (5/2)).