# Diameter of N-Cube/Corollary

Jump to navigation
Jump to search

This article needs to be linked to other articles.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{MissingLinks}}` from the code. |

## Corollary to Diameter of N-Cube

Let $Q_n = \closedint {c - R} {c + R}^n$ be an $n$-cube in Euclidean $n$-Space equipped with the usual metric.

The diameter of $Q_n$ is the length of some diagonal of $Q_n$.

## Proof

To minimize the sum in question, we chose each coordinate $y_i$, $x_i$ of $x$ and $y$ to be endpoints.

Thus any $x, y$ so chosen is a vertex, by the definition of vertex.

Certainly $x \ne y$ because were they equal, the distance between them would be zero, and the sum would not be maximal.

The result follows from the definition of a diagonal.

$\blacksquare$