The generalized Baues problem for cyclic polytopes. I
The cyclic polytope \(C(n,d)\) is the convex hull of any \(n\) pairwise distinct points on the moment curve \(\{(t,t^{2},\ldots ,t^{d}):t\in \mathbb{R\}}\) in \(\mathbb{R}^{d}\). The main result of the paper says that the order complex of the poset of all proper polyhedral subdivisions of \(C(n,d)\), partially ordered by refinement, is homotopy equivalent to an \((n-d-2)\)-sphere. This yields an affirmative answer to the so-called generalized Baues problem for the special case of the vertices of a cyclic polytope. The proof works also for any polytope whose vertices have the alternating oriented matroid, but not for every polytope combinatorially equivalent to a cyclic polytope. As a second result, it is shown that there are four triangulations of \( C(11,5)\) with only four bistellar flips, while the dimension of the secondary polytope is five. One of these triangulations, found by a computer program, is given explicitly. As a consequence, it can be seen that there are triangulations of cyclic polytopes that are non-regular for every choice of points on the moment curve. There is a third result which says that every triangulation of \(C(n,d)\) is a lifting triangulation. This provides a remarkable contrast to the second result, since lifting triangulations are somewhat more general combinatorial analogues of regular triangulations. (A triangulation \(\Delta \) of a polytope \(P\) is called regular if there exists a piecewise linear function on \(P\) which is given by a different linear function on each maximal cell of \(\Delta \). The vertices of the secondary polytope are in one-to-one correspondence with the regular triangulations).
scientific article; zbMATH DE number 1406279
scientific article