Subboxes

Subboxes are axis-aligned hyperrectangles contained within a larger axis-aligned hyperrectangle, or box. If the outer box B is the Cartesian product of intervals B = ∏_{i=1}^n [L_i, U_i], a subbox S is any S = ∏_{i=1}^n [l_i, u_i] with L_i ≤ l_i ≤ u_i ≤ U_i for all i.

Subboxes can be used to partition a domain into smaller regions or to tighten variable bounds. They preserve axis alignment and their definition provides a straightforward way to describe containment relationships between regions in multi-dimensional spaces.

Construction and properties: Subboxes are often obtained by subdividing one or more coordinate directions, such as halving an interval along a coordinate to create two smaller subboxes. They retain axis alignment and their intersection is either a subbox or empty; the union of disjoint subboxes under a partition can cover the parent box. Subboxes form a natural partial order under inclusion.

Applications: In interval arithmetic and guaranteed numerical computation, subboxes represent tighter enclosures of quantities. In constraint satisfaction and global optimization, they form the nodes of a branch-and-bound search tree, where the objective is evaluated over subboxes. In numerical integration and finite element methods, subboxes serve as subdomains for quadrature, error control, and adaptive mesh refinement.

Variations: In higher dimensions, the number of possible subboxes grows with the number of subdivisions. Subboxes are closely related to terms such as hyperrectangles, boxes, and intervals.