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Here is a one-dimensional analogue of Richard's triangle example, obtaining a counterexample in the set of reals. Namely, replace every integer $n$ with two numbers at fixed small distance . For example, let $X$ have $n$ and $n+\frac14$ for every integer $n$. n\pm\epsilon$. One can suitably translate and reflect to realize homogeneity, but there is no isometry swapping $0$ \epsilon$ and $1$.1+\epsilon$. |
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Here is a one-dimensional analogue of Richard's triangle example, obtaining a counterexample in the set of reals. Namely, replace every integer with two numbers at fixed small distance. For example, let $X$ have $n$ and $n+\frac14$ for every integer $n$. One can suitably translate and reflect to realize homogeneity, but there is no isometry swapping $0$ and $1$. |
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