I am trying to show that: Prove or disprove: An unbounded n-dimensional polyhedral set can have at most n extreme directions.
My attempt: I try to create some counter-examples with $n=2$ and $n=3$ cases, however these do not work out well. In each of the cases, we have at most $n$ extreme directions. Also, I try to draw some graphs for $n=2$ case, but it did not work either:(
PS: I try to use the following theorem from the lecture notes: Let $K = \{x \in \mathbb{R}^n: Bx \le 0\}$ be a pointed nontrivial polyhedral cone. A nonzero vector $d \in K$ generates an extreme ray of $K$ if and only if among the inequalities $b_i^Tx \le 0$ for $i=1,2,...k$ defining $K$ there are at least $n-1$ linearly independent inequalities which are active at $d$.
Can anyone give any hints/ideas,please?
Thank you very much, in advance:)