Hypersimplicial subdivisions

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