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?
It follows from 1.6.15(1) in "A Course in Metric Geometry" by Burago, Burago, and Ivanov and 1.6.15(2) is a more general statement:



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> Any distance-noncontracting map from a compact metric space to itself is an isometry.
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This statement is needed in the proof that Gromov--Hausdorff metric is a metric.
Also check the solution of 1.12 in my ["Pure metric geometry"](https://arxiv.org/abs/2007.09846) it is based on a different idea.


