I am currently reading a book on analysis. I have some finite dimensional Hilbert space $H$ and let $B$ be its unit ball. Then it says that the distance of some random point $x \in H$ to the unit ball is given by $$ d(x,B) = \max\{0, 1 - \| x \| \}. $$ But shouldn't it be $$ d(x,B) = \max\{0, \| x \| - 1 \}? $$ If $S_H$ is the unit sphere, then I have $d(x,S_H) = \max\{0,1 - \| x \|, \| x \| - 1 \}$.
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1I agree with you. – nicomezi Aug 27 '18 at 11:19
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you don't need a 0 in your last expression, but yes, the book most likely assumes that the point is inside the ball for whatever reason – Aleksejs Fomins Aug 27 '18 at 11:21
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When the point is inside of the unit sphere $S_H$, then yes, the distance is given by $d(x,S_H)=\max\{0,1-\|x\|\}$. When the point is outside the unit sphere we have $d(x,S_H)=\max\{0,\|x\|-1\}$ as you've said.
Of course that means you are correct, since you are dealing with the unit ball, in which being inside the unit sphere gives us $d(x,B)=0$.
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