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 $f_0(P)+f_0(Q)-(d+1)$ vertices. We show that if for some $1\le i\le d-1$, $P$ 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 $2\le i\le d-2\le 6$ 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 $2\le i\le 4$, there is an infinite family of self-dual $i$-simplicial $i$-simple $2i$-polytopes. Finally, we show that for every $d\ge 4$, there are $2^{\mathrm{\Omega }(N)}$ 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