Let $S\subset\mathbb{R}^n$ be any set. A convex cone $C$ with apex $a$ and non-empty interior is a support cone of $S$ at $a$ if
i) $a \in S,$
ii) $S \subset (\text{int} \;C)^{\complement}$ (i.e complement of $(\text{int} \;C)$) and,
iii) $C$ is a maximal (with respect to inclusion) convex cone with these properties.
My question is, will $(\text{int} \;C)^{\complement}$ be a closed half space or the kernel of a linear functional $f:\mathbb{R}^n\longrightarrow \mathbb{R}$ like that of a support plane?
A text reference will also be really appreciated.
This will not always be true; see this counterexample.
For the open, non-convex (yellow) "pac-man" set $S$, we see that the (red) open support cone emanating from its "mouth" is not a halfspace. Furthermore, since it is a nonlinear set, it cannot be the kernel of a linear functional.
For good references on convex cones, I'd recommend Rockafellar's book on Convex analysis, as well as Bauschke/Combettes' book on Convex Analysis and Monotone Operator theory (2nd edition, Chapter 6).