closedrectangular

Closedrectangular is commonly used to denote the Cartesian product of two closed intervals [a,b] and [c,d] in the Euclidean plane, forming a closed rectangle. This region includes its boundary and is one of the simplest nonempty compact subsets of R^2. When the sides are parallel to the coordinate axes, the term is often described as an axis-aligned closed rectangle; more general rectangles may be obtained by applying a rigid motion (translation and rotation) to an axis-aligned rectangle.

Key properties include that a closed rectangle is closed and bounded, hence compact in R^2 by the Heine–Borel theorem. It is a convex set, meaning any line segment joining two points in the rectangle lies entirely within the rectangle. The boundary consists of four line segments, forming the edges, and the four corner points correspond to the Cartesian product of the interval endpoints: (a,c), (a,d), (b,c), and (b,d). Its area equals (b−a)(d−c) and its perimeter is 2[(b−a)+(d−c)]. In higher dimensions, a closed rectangular box or hyperrectangle is the product [a1,b1]×…×[an,bn], sharing similar properties.

Variants include the open rectangle (a,b)×(c,d), the half-open rectangle, and other grid-aligned regions used in analysis, integration, and computational geometry. Closedrectangular regions frequently serve as simple domains of integration and common bounding objects in algorithms and geometric constructions.