A scheme for computing minimum covers within simple regions

Let X be a simple region (e.g., a simple polygon), and let Q be a set of points in X. Let O be a convex object, such as a disk, a square, or an equilateral triangle. We present a scheme for computing a minimum cover of Q, consisting of homothets of O contained in X. In particular, a minimum disk cover of Q with respect to X, can be computed in polynomial time.