  Let $\mathcal{F}$ be a family of subsets of a ground set $\{1,\ldots,n\}$
with $|\mathcal{F}|=m$, and let $\mathcal{F}^{\updownarrow}$ denote the family
of all subsets of $\{1,\ldots,n\}$ that are subsets or supersets of sets in
$\mathcal{F}$. Here we determine the minimum value that
$|\mathcal{F}^{\updownarrow}|$ can attain as a function of $n$ and $m$. This
can be thought of as a `two-sided' Kruskal-Katona style result. It also gives a
solution to the isoperimetric problem on the graph whose vertices are the
subsets of $\{1,\ldots,n\}$ and in which two vertices are adjacent if one is a
subset of the other. This graph is a supergraph of the $n$-dimensional
hypercube and we note some similarities between our results and Harper's
theorem, which solves the isoperimetric problem for hypercubes. In particular,
analogously to Harper's theorem, we show there is a total ordering of the
subsets of $\{1,\ldots,n\}$ such that, for each initial segment $\mathcal{F}$
of this ordering, $\mathcal{F}^{\updownarrow}$ has the minimum possible size.
Our results also answer a question that arises naturally out of work of Gerbner
et al. on cross-Sperner families and allow us to strengthen one of their main
results.
