metric projection onto one dimensional (closed ) subspace in $L_p (p\neq 2)$

Question

I want to know "if the metric projection onto one dimensional (closed) subspace in $L_p (p\neq 2)$ is linear? I think it is not linear, but I can not give a strict proof.

Thanks for any answer!

Answer 1

No. Consider space $\mathbb{R}_1^2$ and its one dimensional subspace $X=\{(t,t): t\in \mathbb{R}\}$. Then for all $a\in \mathbb{R}$ the distance from $(a,0)$ to $X$ is given by $$ d((0,a), X) =\min_{x\in X} d((a,0), x) =\min_{t\in\mathbb{R}}d((a,0),(t,t)) =\min_{t\in\mathbb{R}}(|a-t|+|t|) $$ You can see that this function is minimized when $t\in[0,a]$. Hence solution of this minimization problem is not unique. So you can not even define metric projection, not to mention its linearity.