Diameter of N-Cube/Corollary

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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.

$\blacksquare$