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I have unit balls defined by the $1$, $2$ and $\infty$ norm in $\mathbb{R}^2$. I want to find the orthogonal projection of a vector $(x,y)$ onto the balls.

How could it be done? I only know how to project vectors orthogonally but have no idea how to do it over the balls.

Thanks a lot.

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    @Siminore, Yup this is correct!2012-05-21

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Recall that the orthogonal projection of a vector $u$ on a vector subspace $V$ is defined as the unique vector $v$ such that (1) $v$ is in $V$, and (2) $u-v$ is orthogonal to $V$, that is, $\langle u,w\rangle=\langle v,w\rangle$ for every $w$ in $V$.

What could be the orthogonal projection of $u$ on a sphere $S$? (In the question you write ball but in the comments it seems clear you mean the unit sphere for a given norm.) Following the classical definition, one should look for $s$ such that (1) $s$ is in $S$ and (2) $u-s$ is orthogonal to... what exactly? Orthogonal to $S$? Alas, no vector $r$ is orthogonal to $S$ in the sense that $\langle r,t\rangle=0$ for every $t$ in $S$, except the null vector. (For example, $t=\langle r,r\rangle^{-1/2}r$ is in $S$ and $\langle r,t\rangle=\langle r,r\rangle^{1/2}\ne0$ for every $r\ne0$.)

This remark shows that you really need to clarify the question.