Consider an arbitrary simplex $\mathcal{S} \subseteq \mathbb{R}^n$ ($\mathcal{S}$ is a polytope in $\mathbb{R}^n$ with $n+1$ vertices and non-empty interior). Let ${\bf P} \in \mathbb{R}^{m \times n}, m<n$ be an orthogonal matrix (in the sense that ${\bf P} {\bf P}^T = {\bf I}$, where ${\bf I} \in \mathbb{R}^{m \times m}$ is the identity matrix) defining a projection over a subspace of dimension $m$ of $\mathbb{R}^n$. Let $\mathcal{P} = {\bf P} \mathcal{S}$ be the projection of the simplex over such subspace.

My question is: what is the maximum amounts of vertices of $\mathcal{P}$?. I am specially interested in low-dimensional cases ($m = 2$, $3$ or $4$ for example). I am also interested in special cases like the maximum amount for regular simplices.

Thanks in advance.

Consider an arbitrary simplex $\mathcal{S} \subseteq \mathbb{R}^n$ ($\mathcal{S}$ is a polytope in $\mathbb{R}^n$ with $n+1$ vertices and non-empty interior). Let ${\bf P} \in \mathbb{R}^{m \times n}, m<n$ be an orthogonal matrix (in the sense that ${\bf P} {\bf P}^T = {\bf I}$, where ${\bf I} \in \mathbb{R}^{m \times m}$ is the identity matrix) defining a projection over a subspace of dimension $m$ of $\mathbb{R}^n$. Let $\mathcal{P} = {\bf P} \mathcal{S}$ be the projection of the simplex over such subspace.

My question is: what is the maximum amounts of vertices of $\mathcal{P}$?. 

I am specially interested in low-dimensional cases ($m = 2$, $3$ or $4$ for example). I am also interested in special cases like the maximum amount for regular simplices.

Thanks in advance.