The merging operation and $(d-i)$-simplicial $i$-simple $d$-polytopes

Abstract

We define a certain merging operation that given two $d$-polytopes $P$ and $Q$ such that $P$ has a simplex facet $F$ and $Q$ has a simple vertex $v$ produces a new $d$-polytope $P\hspace{0.1em}\triangleright Q$ 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 $P\hspace{0.1em}\triangleright Q$. 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 any $d\geq 4$, there are $2^{\Omega(N)}$ combinatorial types of $(d-2)$-simplicial $2$-simple $d$-polytopes with at most $N$ vertices.