Show that each edge of the cyclic polytope $C_4(6)$ is contained in either three or four facets, and either three or four 2-faces.

Question

Note: here $C_4(6)$ is the notation for the cyclic polytope of dimension 4 and of 6 vertices.

By the 2-neighbourly property of $C_4(6)$ and the Dehn-Sommerville equations, I've determined that the polytope has 6 vertices, 15 edges, 18 2-faces and 9 3-faces. Also, since cyclic polytopes are simplicial, their k-faces have exactly (k+1) vertices.

However, these are not enough to answer the questions above. What other characterization of cyclic polytopes that could help me with those questions?

Answer 1

Since the dimension 4 is even, each vertex figure of $C_4(6)$ is combinatorially equivalent to $C_3(5)$, a triangular bipyramid. (You can find a proof of this statement in Facets and Nonfacets of Convex Polytopes by Perles and Shephard, statement (6), on page 117.)

So, considering the vertex figure at a vertex $v$ of $C_4(6)$, there are two vertices of valence three (corresponding to edges incident to $v$ contained in three 2-faces and three facets), and three vertices of valence 4 (corresponding to edges incident to $v$ contained in four 2-faces and four facets).

Moreover, we can count that there are $6 \cdot 2 / 2 = 6$ edges of the first type, and $6 \cdot 3 / 2 = 9$ edges of the second type.