  Our basic result, an isoperimetric inequality for Hamming cube $Q_n$, can be
written: \[ \int h_A^\beta d\mu \ge 2 \mu(A)(1-\mu(A)). \] Here $\mu$ is
uniform measure on $V=\{0,1\}^n$ ($=V(Q_n)$); $\beta=\log_2(3/2)$; and, for
$S\subseteq V$ and $x\in V$, \[ h_S(x) = \begin{cases} d_{V \setminus S}(x)
&\mbox{ if } x \in S, 0 &\mbox{ if } x \notin S \end{cases} \] (where $d_T(x)$
is the number of neighbors of $x$ in $T$).
  This implies inequalities involving mixtures of edge and vertex boundaries,
with related stability results, and suggests some more general possibilities.
One application, a stability result for the set of edges connecting two
disjoint subsets of $V$ of size roughly $|V|/2$, is a key step in showing that
the number of maximal independent sets in $Q_n$ is $(1+o(1))2n\exp_2[2^{n-2}]$.
This asymptotic statement, whose proof will appear separately, was the original
motivation for the present work.
