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.
$\blacksquare$