Are all pointed cones convex?

Question

If a cone is pointed, does that imply it is convex? It feels like it is true, but I want to be sure, since I can't seem to find it outright stated anywhere. For a cone $K$, if $\forall x \neq 0 \in K$, $-x \notin K$, then the cone must be restricted to half of the orthants in whatever dimension we are in. Based on the structure of cones it feels true, based on how cones look like. This is a far cry from a formal proof, hence why I'm concerned it isn't true.

Certainly the inverse isn't true. $R^n$ is a convex cone, but isn't pointed.

Answer 1

Hint: think of a two-scoop ice-cream cone.

Answer 2

No: take a small-enough non-convex planar figure, imbed it in a hyperplane $x+y+z=c$ with $c$ large enough so that the imbedded figure is entirely in the first orthant. Then take all positive scalar multiples of the points in the imbedded image. Pointed, not convex.