I found a statement involving a homeomorphism $f:X\to X$ of a compact metric space $X$, with Lipshitz coefficient 1, i.e., a non-expansive map, and cannot think of an example where $f$ is not an isometry. Must it be?
Is every 1-Lipschitz homeomorphism $f:X\to X$ from a compact metric space to itself an isometry?
Saúl RM
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