Wouldn't an element of a set always spans a ray of the cone of that set?

Question

For context, I'm reading the paper A linear optimization oracle for zonotrope computation.

In algorithm 1, line 8, I don't understand the testing condition:

For a discrete set $G \subset \mathbb{R}^n$.
It is testing condition: $\forall z \in G$ if $z$ spans a ray of $\operatorname{cone}(G)$.

Isn't it always true for all elements of $G$?

Answer 1

Presumably, this paper is using "ray" to mean "extreme ray", that is (borrowing a definition from Fenchel 1951)

A ray (x) of a convex cone C is an extreme ray of C if x is not a positive linear combination of two linearly independent vectors of C.

It seems clear this is what is meant in your paper, given this statement (part of proposition 2.4):

The half-line spanned by a point $x$ in $A$ is a ray of the cone spanned by $A$ if and only if the line through 0 and $x$ is disjoint from the convex hull of $A\setminus\{x\}$.