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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 $k<n$).

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View the two examples, I think $P(n,k)$ is the $(n-k)$-rectified $n$-hypercube or the $(k-1)$-rectified $n$-cross-polytope (same thing). I believe the notion of rectification will be very helpful for answering your other questions.

Properties of such polytpopes when $n$ and $k$ are small can be found on wikipedia. For example http://en.wikipedia.org/wiki/Rectified_10-cubes

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  • $\begingroup$ Thanks a lot. Do you know a good reference where the process of rectification of a polytope is discussed? $\endgroup$
    – Lucien
    Commented Oct 9, 2014 at 11:02

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