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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.

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The cyclic polytope C(4,6) has nine tetrahedron facets. Each of these facets can only border four other facets, so not every facet borders all other facets.

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