  Let $\pi:{\mathbb R}^n \to {\mathbb R}^d$ be any linear projection, let $A$
be the image of the standard basis. Motivated by Postnikov's study of postitive
Grassmannians via plabic graphs and Galashin's connection of plabic graphs to
slices of zonotopal tilings of 3-dimensional cyclic zonotopes, we study the
poset of subdivisions induced by the restriction of $\pi$ to the $k$-th
hypersimplex, for $k=1,\dots,n-1$. We show that:
  - For arbitrary $A$ and for $k\le d+1$, the corresponding fiber polytope
$\mathcal F^{(k)}(A)$ is normally isomorphic to the Minkowski sum of the
secondary polytopes of all subsets of $A$ of size $\max\{d+2,n-k+1\}$.
  - When $A={\mathbf P}_n$ is the vertex set of an $n$-gon, we answer the Baues
question in the positive: the inclusion of the poset of $\pi$-coherent
subdivisions into the poset of all $\pi$-induced subdivisions is a homotopy
equivalence.
  - When $A=\mathbf{C}(n,d)$ is the vertex set of a cyclic $d$-polytope with
$d$ odd and any $n \ge d+3$, there are non-lifting (and even more so,
non-separated) $\pi$-induced subdivisions for $k=2$.
