  We give the structure of discrete two-dimensional finite sets $A,\,B\subseteq
\R^2$ which are extremal for the recently obtained inequality $|A+B|\ge
(\frac{|A|}{m}+\frac{|B|}{n}-1)(m+n-1)$, where $m$ and $n$ are the minimum
number of parallel lines covering $A$ and $B$ respectively. Via compression
techniques, the above bound also holds when $m$ is the maximal number of points
of $A$ contained in one of the parallel lines covering $A$ and $n$ is the
maximal number of points of $B$ contained in one of the parallel lines covering
$B$. When $m,\,n\geq 2$, we are able to characterize the case of equality in
this bound as well. We also give the structure of extremal sets in the plane
for the projection version of Bonnesen's sharpening of the Brunn-Minkowski
inequality: $\mu (A+B)\ge (\mu(A)/m+\mu(B)/n)(m+n)$, where $m$ and $n$ are the
lengths of the projections of $A$ and $B$ onto a line.
