What is an example of non-convex cone

Question

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.

Answer 1

$$\bigvee$$

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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.

Answer 2

The union of the 1st and the 3rd quadrants is a cone but not convex; the 1st quadrant itself is a convex cone.