  Given a combinatorial $(d-1)$-sphere $S$, to construct a combinatorial
$d$-sphere $S^{\hspace{.2mm}\prime}$ containing $S$, one usually needs some
more vertices. Here we consider the question whether we can do one such
construction without the help of any additional vertices. We show that this
question has affirmative answer when $S$ is a flag sphere, a stacked sphere or
a join of spheres. We also consider the question whether we can construct an
$n$-vertex combinatorial $d$-sphere containing a given $n$-vertex combinatorial
$d$-ball.
