There is one detail about hyperplanes I just cant prove myself (it's so simple a direct proof should work)...
Claim:
Given a hyperplane with (non normalized) normal vector $w$, $$ w\cdot x - b =0 , b\in\mathbb{R}, x,w\in\mathbb{R}^n. $$ Then the offset from the hyperplane to the origin in the direction of $w$ is given by $b/\|w\|$.
Proof:
(Thank you AndreaS.)
As it's the offset of the hyperplane to the origin, we should be able to shift all $x$ in the hyperplane by $\frac{b}{\|w\|}$ units along the unit vector $\color{red}{\frac{w}{\|w\|}}$, and have the origin $0$ satisfy the resulting equality. Indeed, the shifted hyperplane becomes $$ w \cdot \left(x+\frac{b}{\|w\|} \color{red}{\frac{w}{\|w\|}}\right) -b =0 $$ $$ w \cdot x + \frac{b( w \cdot w)}{\|w\|^2} -b=0$$ $$ w \cdot x=0$$ We see that $x=0$ satisfies this equality as desired