Covering points with minimum/maximum area orthogonally convex polygons

In this paper, we address the problem of covering a given set of points on the plane with minimum and/or maximum area orthogonally convex polygons. It is known that the number of possible orthogonally convex polygon covers can be exponential in the number of input points. We propose, for the first time, an O(n²) algorithm to construct either the maximum or the minimum area orthogonally convex polygon if it exists, else report the non-existence in O(n log n).