Solution
= Solution
A <down-set> in the <Boolean lattice> is a family closed under taking subsets. Restricting the minimization to down-sets cannot decrease the minimum from the <edge-isoperimetric inequality in the discrete cube>.
The coordinate subcube consisting of all subsets of a fixed $k$-element set is itself a down-set and attains that minimum. Thus
$$
\boxed{g(2^k)=2^k(n-k)}.
$$