Cyclic polytope

A cyclic polytope is a convex polytope with vertices on the moment curve. It is for many issues of combinatorial theory of polytopes of great importance, among others, the Upper - Bound Theorem.

Definition

Be the torque curve in the dimension d

Then the cyclic polytope is the convex hull of n points on the moment curve, where n must be at least as large as d 1.

It is also possible to define the cyclic polytope defined on different torque curves.

Examples

In the two-dimensional, the torque curve with the standard parabola is the same. Each polygon whose vertices lie on the standard parabola is a cyclic polytope.

Properties

• Two gleichdimensionale cyclic polytopes with the same number of corners are combinatorially equivalent. One can therefore speak of the cyclic d- polytope with n vertices. This property follows from the Geradheitskriterium by Gale.
• The cyclic polytope is a simplicial polytope, ie any real side of him is a simplex.
• Furthermore, a - neighborly polytope. Each convex hull of an arbitrary set of vertices is a site in the polytope.
• The outstanding feature of the cyclic polytope is its " extremality ". Among all the d-dimensional polytopes with n vertices, the maximum number of k-dimensional sides ( k < d). A d- polytope with n vertices can therefore no longer k-faces than the corresponding cyclic polytope with n vertices ( Upper -bound theorem).