This question is related to research I am doing on intersection graphs of convex polytopes in Euclidean space.
The following definition comes from this paper:
For any polytope $P$, let $\dim P$ denote its dimension. Let $\mathcal L^d$ be the set of all $(d-1)$-dimensional hyperplanes in $\mathbb R^d$, containing the origin. By ${\mathcal L^d \choose k}$ we denote the set of all $k$-element subsets of $\mathcal L^d$. For any $L \in {\mathcal L^d \choose k}$, by $\mathcal P^d (L)$ we denote the set of all polytopes in $\mathbb R^d$ whose every facet is parallel to one of the hyperplanes in $L$.
Given a regular hyperoctahedron $H_d$ in $\mathbb R^d$, what is the smallest $k$ such that $H$ belongs to $\mathcal P^d (L)$ for some $L \in {\mathcal L^d \choose k}$?
Looking at small examples like $d=2,3$, we have that $H_2$ is a square, so in that case $k=2$ (the facets of $H_2$ are parallel to $2$ lines). For $H_3$, the upper half has $4$ faces parallel to $4$ planes in $\mathbb R^3$, but by symmetry the faces of the bottom half are also parallel to those $4$, so $k=4$.
In general, $H_d$ has $2^d$ facets, but by symmetry I think the answer should be $k=2^d /2$. When $d=2,3$ this checks out.
Is my reasoning correct?
The hyperoctahedron is the convex hull of the $2d$ points with all but one coordinate $0$, the other $\pm 1$.
The facets are the hyperplanes $$ \sum_{i=1}^d \epsilon_ix_i = \pm 1 $$ where each $\epsilon_1 = \pm 1$.
For each of the $2^d$ ways to choose those signs the hyperplanes where the sum is $\pm 1$ are parallel, so there are $2^{d-1}$ parallel pairs.
PS I think this may answer your previous question at Why are balls in the taxicab metric always hyperoctahedrons?