  Let $\Delta$ be a $d$-dimensional normal pseudomanifold, $d \ge 3.$ A
relative lower bound for the number of edges in $\Delta$ is that $g_2$ of
$\Delta$ is at least $g_2$ of the link of any vertex. When this inequality is
sharp $\Delta$ has relatively minimal $g_2$. For example, whenever the
one-skeleton of $\Delta$ equals the one-skeleton of the star of a vertex, then
$\Delta$ has relatively minimal $g_2.$ Subdividing a facet in such an example
also gives a complex with relatively minimal $g_2.$ We prove that in dimension
three these are the only examples. As an application we determine the
combinatorial and topological type of $3$-dimensional $\Delta$ with relatively
minimal $g_2$ whenever $\Delta$ has two or fewer singularities. The topological
type of any such complex is a pseudocompression body, a pseudomanifold version
of a compression body.
  Complete combinatorial descriptions of $\Delta$ with $g_2(\Delta) \le 2$ are
due to Kalai [12] $(g_2=0)$, Nevo and Novinsky [13] $(g_2=1)$ and Zheng [21]
$(g_2=2).$ In all three cases $\Delta$ is the boundary of a simplicial
polytope. Zheng observed that for all $d \ge 0$ there are triangulations of
$S^d \ast \mathbb{RP}^2$ with $g_2=3.$ She asked if this is the only
nonspherical topology possible for $g_2(\Delta)=3.$ As another application of
relatively minimal $g_2$ we give an affirmative answer when $\Delta$ is
$3$-dimensional.
