Can a line be a cone or not?

Question

Can we say that a line is a cone? That is, if I am given set $S$, containing all the points on a line, then that must be cone because for a cone, we require $x \in S$. Then, $cx$ must also be in the set $c \geq 0$

Answer 1

There are many similar definitions for cones. One of the definitions is: The set A is a cone iff i) for any $c>=0$ and for any x in A, cx is also in A, and ii) {x,-x} is a subset of A iff x=0.

For your question I can say that the ray with vertex to zero is a cone (by the definition i've written). The line is not a cone, because it has both positive and negative elements.

But if you take the definition with just the first condition, than the line is a cone too.