Consider vectors $w_1, w_2, w_3, w_4 \in \mathbb{R}^{n}$, $n \in \mathbb{Z}_{\geq 1}$.
Assume the following statement. For all $(c_1,c_2) \in (\mathbb{R}_{\geq 0} \times \mathbb{R}_{\geq 0}) \setminus\{(0,0)\}$ there exists $x \in \mathbb{R}^n$ such that
$ \left( c_1 w_1 + c_2 w_2 \right)^\top x < 0 \quad \text{ and } \quad \left( c_1 w_3 + c_2 w_4 \right)^\top x < 0. $
Find a case in which the following statement is false. There exists $y \in \mathbb{R}^n$ such that
$ w_i^\top y < 0 \quad \forall i \in \{1,2,3,4\}. $