Q:

$l^{1}$-distance and projection in Euclidean space

Let us consider Euclidean space $H = R^{n}$ equipped with standard dot product 
$$
<x,y> = \sum_{i=1}^{n}x_{i}y_{i}
$$
and 
corresponding $l^{2}$-norm $||\cdot||_{2}$. Next, let $S$ be a vector subspace of $H$. 
Let $\Pi(h|S)$ be an orthogonal projection of $h\in H$ onto $S$, i.e.
$$
\Pi(h|S) = argmin_{x\in S}||h - x||_{2}
$$
Assume that $g\in S$ is some vector from $S$. Can you give an example of $S$, $g \in S$ and $h \in H$ but $h \notin S$ such that
$$
||h - g||_{1} < ||\Pi(h|S) - g||_{1},
$$
where $||\cdot||_{1}$ is $l^{1}$-norm. 

A:

consider H = $\mathbb{R}^2$ and $S = <(u,v)>.$ $\, (u^2 + v^2 = 1)$
define $h = (a,b)$ and  $g = (u,v)$. then our criterion will be
$$
\|a-u,b-v\|_1 < |au+bv-1|\|u,v\|_1\\i.e.\quad|a-u|+|b-v|<|au+bv-1|(|u|+|v|)
$$ 
By setting $ s = a- u,\,\, t = b - v$, 
$$
\quad|s|+|t|<|su+tv|(|u|+|v|)
$$ 
Now, assume $s = 1$ .  Moreover, since $\, u^2 + v^2 = 1$, set $u = cos\theta $ and $ v = sin\theta$. 
$$
\quad|1|+|t|<|cos\theta+tsin\theta|(|cos\theta|+|sin\theta|)
$$
you can check $(t,\theta) = (7,1(rad))$ satisfies above inequality. 
Equivalently, $g = (u,v) = (0.5403,0.8415)$ and $h = (a,b) = (1.5403,7.8415)$.

