A tight bound for point guards in piecewise convex art galleries

We consider the problem of guarding curvilinear art galleries. A Jordan arc a joining two points, p and q, in the plane is called a convex arc if the closed curve obtained by joining a with the line segment pq encloses a convex set. A piecewise convex art gallery A with n vertices is a simply connected region in the plane whose boundary consists of n convex arcs where A lies on the convex side of each arc. We show that ⌈n2⌉ point guards are always sufficient and sometimes necessary to guard a piecewise convex art gallery with n vertices. We also give a shorter proof for the sufficiency of ⌊2n3⌋ vertex guards, for n⩾2, which was first derived by Karavelas et al.