Call a vertex $A$ convex if the interior angle at the vertex is less than $ 180^\circ .$ Now note that a polygon with $n$-vertices and $n \geq 4$ has the sum of interior angles equal to $(n-2)180^\circ$. Therefore a polygon has at least $3$ convex vertices.
I do not understand why it has at least $3$ such vertices. I believe that the pigeonhole principle can be applied however I do not see how. Can someone explain this?