Looking for some geometric intuition for half spaces in $\mathbb{R}^{d}$

Question

I've encountered the following in a text about linear algebra (The context is machine learning)

Now, I have a clear understanding of the first two objects (A hyperplane and halfspace, both passing through 0). But the generelazation is somewhat puzzling to me. If I had to define a halfspace moved from the origin by some distance $\alpha$ I'd define $$\left\{ x+\alpha w\mid\left\langle w,x\right\rangle \geq0\right\} $$ (Assuming $w$ is a unit vector, maybe this is the issue)

Why was this defined the way it was? and what are $x_{0}$ and $b$ in the picture?

Answer 1

I think the 3rd bullet point should be

$\{x|(w,x-x_0) \ge 0 \}$

which is then, by linearity of the inner product, equivalent to

$\{x|(w,x) \ge (w,x_0) \}$

or

$\{x|(w,x) \ge b \}$

where $b=(w,x_0)$ is $|w|$ times the projection of $x_0$ on $w$. This is describing a half-space through $x_0$ that is bounded by the hyperplane $\{x|(w,x) = b \}$ through $x_0$ perpendicular to $w$.

Answer 2

The point $x_0$ is a point on the hyperplane boundary of the half-space. Essentially, by replacing $x$ with $x - x_0$ in the set definition, this translates the set by $x_0$, so that what was the origin, becomes $x_0$.

The $b$ is defined in the question: $\langle w, x_0 \rangle$. It's just a number so that the half-space can be expressed as solutions to an inequality of the form, $\langle x, w \rangle \ge \alpha$.