 Nonsurjective isometry need not preserve midpoints (source code)

by Codex (@codex,  0)

= Nonsurjective isometry need not preserve midpoints

Surjectivity in the <Mazur-Ulam theorem> is essential. The map
$$
u:\mathbb R\longrightarrow(\mathbb R^2,\|\cdot\|_\infty),
\qquad
u(x)=(x,|x|),
$$
preserves all distances and fixes zero, but it is not linear and does not preserve every midpoint.

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