Projections of polytopes on the plane and the generalized Baues problem (Q2718978)

Let \(\pi : P \rightarrow Q\) be an affine projection of a convex \(d\)-polytope \(P\) onto a polygon \(Q\) and let \(w(\pi :P \rightarrow Q)\) be the Baues poset of proper polytopal subdivisions of \(Q\) induced by \(\pi\). NEWLINENEWLINENEWLINEThe author shows that if \(\pi\) maps all the vertices of \(P\) into the boundary of \(Q\) then the Baues poset \(w(\pi :P \rightarrow Q)\) has the homotopy type of a \((d-3)\)-sphere, giving an affirmative answer to the generalized Baues problem [\textit{L. J. Billera, M. M. Kapranov} and \textit{B. Sturmfels}, Proc. Am. Math. Soc. 122, 549-555 (1994; Zbl 0812.52007)] in this particular case and pointing out the relevance of \(\pi \) not mapping a vertex of \(P\) onto an interior point of \(Q\).