Let $n$ and $k$ be positive integers with $k\leq n$.

Let $P(n,k)$ be the convex hull in $\mathbb R^n$ of the $2^k {n \choose k}$ vectors whose exactly $k$ coordinates belong to $\{\pm 1\}$ all the others being equal to $0$.

Examples: $P(n,n)$ is the $n$-hypercube; $P(n,1)$ is the $n$-cross-polytope.

Viewed the two classical examples above, I expect that the $P(n,k)$'s have been studied already. Some references would be welcome.

I'm interested in the $f$-vector and more specifically to the facets (1-codimensional faces) of $P(n,k)$: how many and what kind of polytopes they are?

Added: thinking to this question, it comes to my mind that $2n$ of the facets of $P(n,k)$'s are $P(n-1,k-1)$ (assuming that $k>1$) and that all the other ones are $(n,k)$-hypersimplexes (assuming that $k<n$).