This might be a very simple question. I am trying to clarify the meaning of extreme direction of closed convex set $S$.
In many books/sources I found, a direction $d$ of $S$ is one such that for any $s\in S$ and scalar $c$, we have $d+cs\in S$. Two directions of $S$ are distinct if one is not a positive multiply of the other. A direction $d$ of $S$ is called extreme if:
(*) it cannot be written as a positive linear combination of two distinct directions, that is,
(**) if $d=c_1d_1+c_2d_2$ with $c_1,c_2>0$, then $d_1=cd_2$ for some $c>0$.
I mark with (*) and (**) because in many sources, either both (*) and (**) are used, or just (*), or just (**).
However, I have some problems grasping this. For example, if some sources only write (**), but it is meant to be equivalent to (*), then isn't it wrong? Shouldn't we have:
(***) if $d=c_1d_1+c_2d_2$ with $c_1,c_2>0$ and $d_1,d_2$ are directions of $S$, then $d_1=cd_2$ for some $c>0$.
which is equivalent to (*)?
If some sources write only (*), then I only see (***) being the equivalent statement.
If some sources write both (*) and (**), then I see two non equivalent statements.
Either my understanding is incorrect, or something else is incorrect?
Thank you very much for the guidance.