Please refer to the photo I have attached to this question.
A set S is convex if and only if for all $\textbf{a},\textbf{b} \in S$, the point $\textbf{x} = c\textbf{a} + d\textbf{b}$ is also in S, provided that $c,d \geq 0$ and $c + d = 1$.
The region displayed in the image attached to this question satisfies this condition, but what's throwing me off is the fact that the region is bounded only on two sides and then extends upward to infinity. Based on the definition of a convex set it seems that this shouldn't matter, because although the region is extending to infinity you would still be able to connect any two points in the region with a line that is also entirely in the region. But perhaps I'm missing something.
Any help would be appreciated. Thanks for your time.
Convex sets need not be bounded. The shaded set is convex. It is in fact a convex cone.