Small-set edge isoperimetry in a square grid ($f_n(a)=2b-1$)

For the $n\times n$ grid graph, if $b$ is an integer, $b<n/2$, and $(b-1{)}^2<a\le b(b-1)$, the minimum edge boundary of an $a$-vertex set is $2b-1$. A corner $(b-1)\times (b-1)$ square with a nonempty partial adjacent column attains it. The lower bound counts occupied rows and columns; full rows or columns are treated separately because they contribute no boundary in their own direction.