That is true, since this is so for $\mathbb RP^n$ - take $n+2$ vertices of a regular simplex in it -- i.e. take the regular simplex in $S^n$ and project its vertices to $\mathbb RP^n$.

That is true, since this is so for $\mathbb RP^n$ - take $n+2$ vertices of a "regular simplex" in it -- i.e. take the regular simplex in $S^n$ and project its vertices to $\mathbb RP^n$.

In coordinates, take points $(1,0,\ldots,0)$, ... $(0,\ldots,0,1)$ in $\mathbb R^{n+2}$, take the hyperplane $\sum_i x_i=1$ and take the $S^n$ in it that passes through these points. Then project this to $\mathbb RP^n$ by the central symmetry of $S^n$

That is true, since this is so for $\mathbb RP^n$ - take the regular $n+2$ simplex in it.

That is true, since this is so for $\mathbb RP^n$ - take $n+2$ vertices of a regular simplex in it -- i.e. take the regular simplex in $S^n$ and project its vertices to $\mathbb RP^n$.