Convex Cones are closed under addition, can we infer from $A + B \in C$ that $A, B \in C$?

Question

Let $C$ be a convex cone, then a defining property is: $A,B\in C\Rightarrow A + B\in C$.

My question (as general as possible) is whether the reverse implication is true, that is: $A + B\in C\Rightarrow A,B\in C$ for a convex cone (will the statement proper cone make any difference)??

If it is true, would anyone care to point in the direction of a proof?

Answer 1

What if your cone is the first quadrant in the plane, $A = (3,-1)$ and $B= (-1,3)$?