Abstract: Let ?:Rn?Rd 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 ? to the k-th hypersimplex, for k=1,?,n?1 . We show that: For arbitrary A and for k?d+1 , the corresponding fiber polytope 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=Pn is the vertex set of an n-gon, we answer the Baues question in the positive: the inclusion of the poset of ? -coherent subdivisions into the poset of all ? -induced subdivisions is a homotopy equivalence. When A=C(d,n) is the vertex set of a cyclic d-polytope with d odd and any n?d+3 , there are non-lifting (and even more so, non-separated) ? -induced subdivisions for k=2 .

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