A remarkable result of I. Shemer [4] states that the combinatorial structure of a neighbourly 2m-polytope determines the combinatorial structure of each of its subpolytopes. From this, it follows that every subpolytope of a cyclic 2m-polytope is cyclic. In this note, we present a direct proof of this consequence that also yields that certain subpolytopes of a cyclic (2m+ 1)-polytope are cyclic.

We introduce the k-stellated spheres and consider the class Wk(d) of triangulated d-manifolds all whose vertex links are k-stellated, and its subclass W∗ k (d) consisting of the (k + 1)-neighbourly members of Wk(d). We introduce the mu-vector of any simplicial complex and show that, in the case of 2-neighbourly simplicial complexes, the mu-vector dominates the vector of its Betti numbers compon...

We introduce a quantitative parameter measuring m-neighbourliness of symmetric convex polytopes in R . We discuss this parameter for random polytopes generated by subgaussian vectors and show its stability properties.