K-cell (mathematics)


A k-cell is a higher-dimensional version of a rectangle or rectangular solid. It is the Cartesian product of k closed intervals on the real line. This means that a k-dimensional rectangular solid has each of its edges equal to one of the closed intervals used in the definition. The k intervals need not be identical. For example, a 2-cell is a rectangle in such that the sides of the rectangles are parallel to the coordinate axes.

Formal definition

Let ai ∈ and bi ∈. If ai < bi for all i = 1,...,k, the set of all points x = in whose coordinates satisfy the inequalities ai ≤ xi ≤ bi is a k-cell.
Every k-cell is compact.

Intuition

A k-cell of dimension k ≤ 3 is especially simple. For example, a 1-cell is simply the interval with a < b. A 2-cell is the rectangle formed by the Cartesian product of two closed intervals, and a 3-cell is a rectangular solid.
The sides and edges of a k-cell need not be equal in length; although the unit cube is a 3-cell, the set of all 3-cells with equal-length edges is a strict subset of the set of all 3-cells.