  Given a set of disjoint simple polygons $\sigma_1, \ldots, \sigma_n$, of
total complexity $N$, consider a convexification process that repeatedly
replaces a polygon by its convex hull, and any two (by now convex) polygons
that intersect by their common convex hull. This process continues until no
pair of polygons intersect.
  We show that this process has a unique output, which is a cover of the input
polygons by a set of disjoint convex polygons, of total minimum area.
Furthermore, we present a near linear time algorithm for computing this
partition. The more general problem of covering a set of $N$ segments (not
necessarily disjoint) by min-area disjoint convex polygons can also be computed
in near linear time.
  A similar result is already known, see the work by Barba et al. [BBB+13].
