I have a convex polytope $P$ in $R^2$ of $\dim P = \dim\operatorname{aff} P = 2$, $\dim\operatorname{aff}P$ is the dimension of the affine hull of $P$. Let $L$ be the line orthogonal to the normal vector $v$. I look at the orthogonal projection of $P$ to $L$, and would like to show that for each inner point $p$ in the projection there are $2$ points on the boundary of $P$ which are projected into $p$.
I hope someone is able to help me give a rigorous proof.