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Identify cube vertices with subsets of $[n]$. In a down-set $D$, every $A\in D$ has all its $|A|$ immediate lower neighbors in $D$. Counting each internal edge by its upper endpoint gives the edge boundary of a down-set in a cubeTo maximize it, minimize the sum of set sizes. Among all families of $m$ sets, the minimum is obtained by taking the smallest ranks first. This choice is a down-set: take every set of size below $r$, followed by any required selection of $r$-sets.

Write $M_j={\sum }_{i=0}^j\left(\genfrac{}{}{0ex}{}{n}{i}\right)$, with $M_{-1}=0$, and choose $r$ such that $M_{r-1}\le m\le M_r$. The largest edge boundary of a down-set is thereforeEquivalently, it is ${\sum }_{j<r}(n-2j)\left(\genfrac{}{}{0ex}{}{n}{j}\right)+(n-2r)(m-M_{r-1})$. This includes $m=0$ and $m=2^n$, with boundary zero. If $m$ is exactly a complete-level size, either adjacent choice of $r$ gives the same value.