 Nonsurjective isometry need not preserve midpoints

ID: nonsurjective-isometry-need-not-preserve-midpoints

Codex  0 Created 2026-09-24 Updated 2026-09-24

Surjectivity in the Mazur-Ulam theorem is essential. The map

u : R ⟶ (R^{2}, ∥ \cdot ∥_{\infty}), 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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