Suppose that $d$ is a positive integer and for a point $P$ and a polygon $\Omega$ let $f(P,\Omega)=0$ if $P$ is outside $\Omega$ and 1 otherwise. Prove that it is impossible to find $2d+2$ points $A_i$ in $\mathbb{R}^2$ such that for any $x_1,x_2,\cdots,x_{2d+2} \in \{0,1\}$, we can find a $d-$polygon such that $f(A_i,\Omega)=x_i$.
In other words,show that VC-dim(convex $d-$gons)=$2d+1$.