discrete geometry - Combinatorial distance ≡ Euclidean distance

I wonder if the class of polytopes I am going to define might have property X:

Consider the regular n-simplex $Delta^n$.

Let $F_k^n$ be the set of k-dimensional faces of $Delta^n$:

• $F_0^n$ = the set of vertices


• $F_1^n$ = the set of edges


• ...


• $F_n^n$ = $Delta^n$

Let $P_k^n$ be the polytope the vertices of which are the centers of the elements of $F_k^n$.

$P_k^n$ represents in a natural way the subsets of [n+1]={0,1,..,n} with exactly k+1 elements.

$P_1^3$ (= $P_2^3$) is the octahedron.

$P_1^4$ (= $P_3^4$) is the rectified 4-simplex (with the triangular prism for vertex figure).

Claim:
$P_1^n$ (= $P_{n-1}^n$) is the rectified n-simplex.

Claim:
For any vertex v of the regular hypercube $C^n$ the vertices with combinatorial distance k to v are the vertices of $P_k^n$.

Conjecture:
*For all n, k, the polytope $P_k^n$ has property X.*

Question: Is there a standard name for the polytopes $P_k^n$?

Question: Can anyone canonically name some other $P_k^n$ for 1 < k < n-1 (like "rectified n-simplex" for k=1)?

Question: Does a proof of the above conjecture seem to be (i) feasible, (ii) trivial, or - if (i) but not (ii) - does anyone (iii) could sketch a proof?

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