Cyclic polytopes and d-order curves (Q1093163)

Let \({\mathbb{R}}^ d\) be a real d-space. Then every hyperplane of \({\mathbb{R}}^ d\) meets a curve of order d in \({\mathbb{R}}^ d\) in at most d points. It is known that the convex hull of n distinct points of a curve of order d is a cyclic polytope. In the case that d is even, the author proves the following conjecture of \(M.\quad A.\quad Perles:\) The set of vertices of a cyclic d-polytope can be extended to a curve of order d.