Combinatorial Theory

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 \(P \triangleright Q\) with \(f_0(P)+f_0(Q)-(d+1)\) vertices. We show that if for some \(1\leq i\leq d-1\), \(P\) and \(Q\) are \((d-i)\)-simplicial \(i\)-simple \(d\)-polytopes, then so is \(P \triangleright Q\). We then use this operation to construct new families of \((d-i)\)-simplicial \(i\)-simple \(d\)-polytopes. Specifically, we prove that for all \(2\leq i \leq d-2\leq 6\) 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 \(2\leq i\leq 4\), there is an infinite family of self-dual \(i\)-simplicial \(i\)-simple \(2i\)-polytopes. Finally, we show that for every \(d\geq 4\), there are \(2^{\Omega(N)}\) 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