Coverings

Given a set $S$ in the plane (or space), a covering of $S$ is a collection of sets (often simple shapes like rectangles, disks, or polygons) whose union contains $S$. That is, every point in $S$ is inside at least one of the covering sets.

Important properties

• Coverings may overlap: the covering sets do not need to be disjoint.

• A covering is different from a partition or tiling, where the pieces usually do not overlap and exactly fill the space.

• The goal in problems may be to use as few covering sets as possible, or to minimize their total area.