for arbitrary dimension, what are the convex polytopes such that all vertices share a facet of some dimension, which is not the top facet (the entire polytope), with all other vertices? One example is simplexes of any dimension, as all vertices share an edge with all other vertices. Are there other types of convex polytope with this property of shared facets?
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Yes, there are. For example, take a k-fold pyramid over a square ($k \ge 1$). If you choose 2 vertices from the square, they share the square. If you take a vertex from the square and an apex, they share an edge. If you take 2 apexes, they share an edge.
