Not all cones is convex cones, because convex cones need set C is convex. Examply: y=|x|, two point (-2,1);(1,1) in C,the sum is (-1,3), not in C, so not convex. but y>=|x| is convex cones.
Note: Convex Optimization P24: Every affine set is also convex, since it contains the entire line between any two distinct points in it, and therefore also the line segment between the point.
because convex set only required theta in [0,1], however, affine set also required theta in(-infty,0) and (0,+infth). So affine set is more demanding, more specific.