Let $(X,d_{X})$ be a distance space and $f,g\colon X\longrightarrow X$ non-constant maps such that $g^{2}=g,f^{2}=f$. Assume that $$ d_{X}(fg(x),fg(y))\leq d_{X}(f(x),f(y))\leq d_{X}(x,y) $$for any $x,y\in X$. Then, is $g$ a distance-decreasing map? For example let $X=\mathbb{R}$ and $f=s\times$ for any $s\in \mathbb{R}^{\times}$ with $|s|<1$, then $g$ is distance-decreasing. I can't immediately think of a counterexample.
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Let $f$ be a constant map, then the first two terms are always zero, independent of $g$.
Candyblock
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Your answer could be improved with additional supporting information. Please [edit] to add further details, such as citations or documentation, so that others can confirm that your answer is correct. You can find more information on how to write good answers in the help center. – Community Oct 20 '22 at 13:26
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3(+1) Please don't obey that bot. Don't damage your answer. – Anne Bauval Oct 20 '22 at 13:30
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Thanks. However, I am assuming a situation where $f$ is not a constant. – M masa Oct 20 '22 at 13:49
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2@Mmasa - Then you need to edit your question to include that assumption. – JonathanZ Oct 20 '22 at 14:04
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I have found a counterexample and will withdraw the question today. – M masa Oct 21 '22 at 04:37