  The first author showed that for a given point $p$ in an $nk$-polytope $P$
there are $n$ points in the $k$-faces of $P$, whose barycenter is $p$. We show
that we can increase the dimension of $P$ by $r$, if we allow $r$ of the points
to be in $(k+1)$-faces. While we can force points with a prescribed barycenter
into faces of dimensions $k$ and $k+1$, we show that the gap in dimensions of
these faces can never exceed one. We also investigate the weighted analogue of
this question, where a convex combination with predetermined coefficients of
$n$ points in $k$-faces of an $nk$-polytope is supposed to equal a given target
point. While weights that are not all equal may be prescribed for certain
values of $n$ and $k$, any coefficient vector that yields a point different
from the barycenter cannot be prescribed for fixed $n$ and sufficiently large
$k$.
