2 problems

Let $2\leq i\leq d/2$. Let $\Delta$ be either the boundary complex of a simplicial $d$-polytope with its natural embedding $p$, or a simplicial $(d-1)$-sphere with a generic embedd…

Let $d\geq 4$, let $2\leq i\leq d/2$, and let $P$ be a simplicial $d$-polytope. Assume that $P$ has no missing faces of dimension at least $d-i+1$. Equivalently, let…