  We study face numbers of simplicial complexes that triangulate manifolds (or
even normal pseudomanifolds) with boundary. Specifically, we establish a sharp
lower bound on the number of interior edges of a simplicial normal
pseudomanifold with boundary in terms of the number of interior vertices and
relative Betti numbers. Moreover, for triangulations of manifolds with boundary
all of whose vertex links have the weak Lefschetz property, we extend this
result to sharp lower bounds on the number of higher-dimensional interior
faces. Along the way we develop a version of Bagchi and Datta's $\sigma$- and
$\mu$-numbers for the case of relative simplicial complexes and prove stronger
versions of the above statements with the Betti numbers replaced by the
$\mu$-numbers. Our results provide natural generalizations of known theorems
and conjectures for closed manifolds and appear to be new even for the case of
a ball.
