I was studying Stephen Boyd's textbook on convex optimization. It says "A set C is called a cone or nonnegative homogeneous, if for every x $\in$ C, we have $\theta x \in $ C. A set C is a convex cone if it is convex and a cone."
I'm just wondering what set could be a cone but not convex.
The union of the 1st and the 3rd quadrants is a cone but not convex; the 1st quadrant itself is a convex cone.
For anyone who came across this in the future.
I was reading that book and was confused for a moment too. A key observation is that the definition is stating the set $C$ is a cone if
$$ \forall x \in C \text{, and } \theta \geq 0, \theta x \in C. $$
This only contains the actual line of that cone; it doesn't contain the interior of the cone. When it is actually convex, it will contain the interior. Geometrically, the V shape is only conic, the V shape with the interior filled in is conic and convex.
$$\bigvee$$
For example, the graph of $y=|x|$ is a cone that is not convex; however, the locus of points $(x,y)$ with $y \ge |x|$ is a convex cone.
In my opinion, the definition of cone applies to any arbitrary set. Therefore we can think about some relatively different cases from what we usually do.
As the figure shown in the link given below, I drew two disconnected curves as the point set C, illustrated by red curves. The cone and convex cone areas are illustrated by yellow shade. In this case, the cone is not convex yet the convex cone is (of course) convex.
