# Cyclic $d$-polytope -- does every facet border every other facet?

A cyclic $$d$$-polytope is "neighborly," where every set of vertices sized $$d/2$$ together make up a face. What I'm interested in: in a cyclic $$d$$-polytope with $$N$$ facet ($$d-1$$ dimensional faces), does every facet "border" every other facet (i.e., share a $$d-2$$ dimensional face)?

Why I think it might: we can use the upper bound theorem in dual form (https://www.cs.mcgill.ca/~fukuda/soft/polyfaq/node12.html) to find the total number of $$d-2$$ dimensional faces for a $$d$$-polytope with $$N$$ facets. If you use the formula in the provided link, we find that there are $$N(N-1)/2$$ such $$d-2$$ dimensional faces. Since every such face borders $$2$$ facets, this total number of $$d-2$$ dimensional faces suggests that the property holds.