How to define a map $f$ from $\mathbb R^n$ to the euclidean unit ball in such a way that $f$ is a contraction? I was thinking about the map that sends $x\to\frac{x}{\|x\|}$, where $\|x\|$ denotes the euclidean norm of $x$, but the image of this map is the sphere $S^{n-1}$, not the whole ball.
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How about $f\equiv 0$? – geetha290krm Feb 05 '24 at 10:00
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1Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. – José Carlos Santos Feb 05 '24 at 10:09
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Can you think of a smooth homeomorphism $f$ of ${\mathbb R}$ to a finite interval such that $|f'(x)|\le 1$ for all $x\in {\mathbb R}$? – Moishe Kohan Feb 05 '24 at 11:03