Construction of the minimal enclosing parallelogram

We consider the problem of constructing a minimal-area parallelogram that contains a given $n$-point set $N$. We give a characterization of locally extremal parallelograms; in the nondegenerate case their number is equal to the number of sides of the convex hull of $N$. This makes it possible to give an algorithm for solving the problem with complexity $O(n\log n)$. We also consider the situation when the convex hull $N$ is known a priori. We indicate a method for transfer from one local extremum to another that makes it possible to lower the estimate of the complexity to $O(n)$.