Maximum-area and maximum-perimeter rectangles in polygons

We study the problem of finding maximum-area and maximum-perimeter rectangles that are inscribed in polygons in the plane. There has been a fair amount of work on this problem when the rectangles have to be axis-aligned or when the polygons are convex. We consider this problem in polygons with n vertices that are not necessarily convex, possibly with holes, and with no restriction on the orientation of the rectangles. We present an algorithm that computes a maximum-area rectangle and a maximum-perimeter rectangle in O(n3log⁡n) time using O(kn2+n) space, where k is the number of reflex vertices of the polygon. Our algorithm can report all maximum-area rectangles in the same time using O(n3) space. We also present a simple algorithm that finds a maximum-area rectangle inscribed in a convex polygon with n vertices in O(n3) time using O(n) space.