We define a certain merging operation that given two d-polytopes P and Q such that P has a simplex facet and Q has a simple vertex produces a new d-polytope P▹Q with f0(P)+f0(Q)−(d+1) vertices. We show that if for some 1≤i≤d−1, P and Q are (d−i)-simplicial i-simple d-polytopes, then so is P▹Q. We then use this operation to construct new families of (d−i)-simplicial i-simple d-polytopes. Specifically, we prove that for all 2≤i≤d−2≤6 with the exception of (i,d)=(3,8) and (5,8), there is an infinite family of (d−i)-simplicial i-simple d-polytopes; furthermore, for all 2≤i≤4, there is an infinite family of self-dual i-simplicial i-simple 2i-polytopes. Finally, we show that for every d≥4, there are 2Ω(N) combinatorial types of (d−2)-simplicial 2-simple d-polytopes with at most N vertices.
Mathematics Subject Classifications: 52B05, 52B11
Keywords: Connected sums, face lattice, face numbers, Gosset-Elte polytopes, self-dual polytopes
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