6
$\begingroup$

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 $k<n$).

$\endgroup$
6
$\begingroup$

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

$\endgroup$
  • $\begingroup$ Thanks a lot. Do you know a good reference where the process of rectification of a polytope is discussed? $\endgroup$ – Lucien from IHP Oct 9 '14 at 11:02

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.