In Boyd's textbook "Convex Optimization" there is an example:
In the above proof that PSD cone is convex, there is this claim that the set of the form $\{X \in \mathbf{S}^n| z^TXz \geq 0\}$ are "halfspaces in $\mathbf{S}^n$".
From my understanding reading from the same text, a hyperplane is the set $\{x \in \mathbb{R}^n | a^Tx = b\}$, and a halfspace is when the set has $\geq$ or $\leq$. This definition only works for set of vectors.
Therefore, how is halfspace in $\mathbf{S}^n$ or a hyperplane in $\mathbf{S}^n$ defined? And do these concepts carry the same graphical intuition as in the vector case?