Half-cube vertex-boundary extrema ($\min |\partial A|=\left(\genfrac{}{}{0ex}{}{n}{\lfloor n/2\rfloor }\right),\max |\partial A|=2^{n-1}$)

For $n\ge 1$, a family of $2^{n-1}$ vertices in the Boolean hypercube has these minimum and maximum external vertex boundary sizes. The Harper theorem gives the minimum: in odd dimension take the lower half of the levels, and in even dimension add the middle-level sets containing coordinate one to all lower levels. The family of even Hamming weight attains the maximum, since every remaining vertex has an adjacent even-weight vertex.