Inverse Additive Problems for Minkowski Sumsets I

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.