VC dimension is defined as the maximum number of points that can be arranged such that classifier $f$ shatters them.
Here we assume the space where the points live is $\mathbb R^n$ and the classier be a hyperplane $\mathbb R^{n-1}$. From my intuition, $VC(\mathbb R^n) = n+1$ where the $n+1$ points should not be in a hyperplane $\mathbb R^{n-1}$, e.g. $VC(\mathbb R^2)=3$ where the 3 points should not be on the same line ($\mathbb R^1$).
If we define the projection $P_-: \mathbb R^{n} \rightarrow \mathbb R^{n-1}$, then $VC(P_- \mathbb R^n) = VC(\mathbb R^{n-1})$.
However, if we define the projection $P: \mathbb R^n \rightarrow \mathbb RP^{n-1}$, then how to define $VC(\mathbb RP^{n-1})$? A hyperplane $\mathbb R^{n-1}$ certainly cannot separate two points in $\mathbb R^n$ along the same ray.