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Let $E$ a normed vector space. Let $B = \{x \in E, ||x|| \leq 1\}$ a closed ball. It's a closed convex.

$E$ is a Hilbert space. I have to determine an expression of the projection on the closed unit ball.

I know that the projection $P_B$ can be written as it : $\forall y \in B, ||x-P_B(x)|| \leq ||x-y||$, ($x \in E$) with $P_B : E \rightarrow B$. But I don't see how to determine the expression of the projection. Someone could help me ? Thank you in advance and I hope my English is understandable.

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1 Answer 1

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For $\|x\|\ge 1:$ $$P_B(x) = \frac{x}{\|x\|}.$$ For $\|x\|\le 1:$ $$P_B(x) = x.$$

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