For any non-zero $\mathbf{y}\in\mathbb{R}^\mathcal{l}$ one half-space through origin is defined by $H_{\mathbf{y}}^{\leq}(\mathbf{0})=\left\{\mathbf{x}\in\mathbb{R}^{\mathcal{l}}:\mathbf{y}\cdot \mathbf{x}\leq\mathbf{0} \right\}$. I want to show $\mathbf{a}\in H^{\leq}_{\mathbf{y}}(0)$ iff $\mathbf{a}=\mathbf{x}-s\mathbf{y}$, where $\mathbf{x}$ is orthogonal to $\mathbf{y}$ and $s\geq 0$.
$\Leftarrow$ side is straightforward, but I couldn't prove $\Rightarrow$ side. I defined $x_i=a_i$ if $ y_i=0$ and $x_i=0$ if $y_i\neq 0$, but couldn't come up with something.Thanks for any help.
$\Rightarrow$ Let it be that $\mathbf{a}\in H_{\mathbf{y}}^{\leq}(\mathbf{0})$ or equivalently $\mathbf{y}.\mathbf{a}\leq0$.
We have $\mathbf{a}=\mathbf{a}-s\mathbf{y}+s\mathbf{y}$ for any scalar $s$.
Now choose $s$ such that $\mathbf{a}-s\mathbf{y}$ and $\mathbf{y}$ are orhogonal.
Then $0\geq\mathbf{y}.\mathbf{a}=s\mathbf{y}.\mathbf{y}$ and consequently $s\geq0$.