Is the orthogonal projection of a bounded point bounded?

Asked Jul 24 '23 at 02:40

Active Jul 24 '23 at 03:21

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Let $v \in \mathbb{R}^n$ be a bounded point ($||v||\leq M$ for some $M>0$) and $C$ be a closed set.

Is $y \in P_C(v)$ a bounded point (it may not be unique when $C$ is not convex) where $P_C(x)=\arg\min_{z\in C} ||z-v||^2$?

My intuition:

I know the orthogonal projection cannot expand things so a bounded point stays bounded but I do not know how to show it. Also, I know the orthogonal projection is continuous (correct me if I am wrong) and I think the image of a bounded point under a continuous map is bounded. If that is the case I am done.

edited Jul 24 '23 at 03:21

asked Jul 24 '23 at 02:40

Saeed

2

What is your definition of "bounded point"? – Greg Martin Jul 24 '23 at 02:48
1

What is $P_C$? Your definition is wrong as it stands. – Ted Shifrin Jul 24 '23 at 03:28
Every point in $\mathbb{R}^n$ is bounded. If the set $C$ is non empty then since $|z-v| \le |z| + |v|$ we see that $|y| \le \inf_{z \in C} |z| + |v|$. – copper.hat Jul 24 '23 at 03:56

Is the orthogonal projection of a bounded point bounded?

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