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 $\mathbb {R} ^{2}$ such that the sides of the rectangles are parallel to the coordinate axes. Every $k$-cell is compact.

Projections of K-cells onto the plane (from $k=1$ to $6$). Only the edges of the higher-dimensional cells are shown.

Formal definition

For every integer $i$ from $1$ to $k$, let $a_{i}$ and $b_{i}$ be real numbers such that for all $a_{i}<b_{i}$. The set of all points $x=(x_{1},\dots ,x_{k})$ in $\mathbb {R} ^{k}$ whose coordinates satisfy the inequalities $a_{i}\leq x_{i}\leq b_{i}$ is a $k$-cell.

Intuition

A $k$-cell of dimension $k\leq 3$ is especially simple. For example, a 1-cell is simply the interval $[a,b]$ 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 (Euclidean) length; although the unit cube (which has boundaries of equal Euclidean length) 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.

References

• Foran, James (1991-01-07). Fundamentals of Real Analysis. CRC Press. ISBN 9780824784539. Retrieved 23 May 2014.
• Rudin, Walter (1976). Principles of Mathematical Analysis. McGraw-Hill.