Cyclic polytope#Gale evenness condition

The Geradheitskriterium by Gale describes a condition on the vertex sets of facets of a cyclic polytope. It goes back to the mathematician David Gale.

One consequence of Geradheitskriteriums of Gale is that two gleichdimensionale cyclic polytopes with the same number of corners are combinatorially equivalent. One can speak of the cyclic d- polytope with n vertices.

Definition

Be the vertex set of a cyclic polytope and had a lot of d vertices of the polytope.

The convex hull of a facet is then if and only if each pair is separated from two corners of an even number of vertices on the moment curve.

Example

Is a cyclic polytope in dimension 3 with 6 corners. Its corners are numbered according to their order on the torque curve. is a set consisting of three vertices of the polytope.

Consists of.

• The edges 1 and 2 are separated by 0 corners on the moment curve.
• 1 and 5, the corners 3 and 4 are separated from the two corners on the moment curve.
• 2 and 5, the corners 3 and 4 are separated from the two corners on the moment curve.

Is therefore a facet of.