hyperintervallet

A hyperintervallet is a small, localized hyperrectangle in d-dimensional Euclidean space, defined as the Cartesian product of closed univariate intervals I_i = [l_i, u_i] for i = 1 to d. It is typically constrained so that its side lengths are small, often below a specified tolerance ε, so that max_i(u_i − l_i) ≤ ε. The term is formed by combining hyperinterval, the axis-aligned bounding box in R^d, with the diminutive suffix -let, indicating a smaller unit within a partitioned structure.

Properties and scope: A hyperintervallet is convex and compact, and its orientation is axis-aligned with respect to the coordinate axes. Its volume equals the product of its side lengths. It is a subset of a hyperrectangle and can be nested or partitioned further in hierarchical schemes. Its primary role is as a basic unit in adaptive partitioning, mesh refinement, and local analysis.

Generation and operations: Hyperintervallets arise from subdividing a larger hyperinterval along coordinate directions, often by bisecting one or more intervals. Common operations include refining a hyperintervallet to create smaller children, computing its center point, and evaluating function bounds over the interval for numerical methods such as interval arithmetic or branch-and-bound algorithms.

Applications: They are used in global optimization, interval analysis, and numerical integration to provide localized bounds, error control, and adaptive discretization. In spatial indexing and computational geometry, hyperintervallets serve as elementary building blocks for partitioning high-dimensional space.

Example: In R^2, a hyperintervallet may be [0, 1/2] × [1/3, 2/3], a small axis-aligned box employed within a partitioning scheme.

See also: hyperrectangle, interval, axis-aligned bounding box.