# What is the lower bound for the number of facets that a general convex $d$-polytope with $n$ vertices can have?

I am familiar with Barnette's Lower Bound Theorem on the number of facets a $$d$$-dimensional simplicial convex polytope with $$n$$ vertices can have. Is there a similar result for a general (i.e. not necessarily simplicial) $$d$$-polytope with $$n$$ vertices?

## 1 Answer

A lower bound on the number of facets can be obtained from McMullen's upper bound theorem. It says that, for a given number of vertices, neighborly polytopes maximize the number of faces in all dimensions (among all polytopes, simplicial or not). In particular, a $$d$$-polytope $$P$$ with $$m$$ vertices has at most $$F(d,m)$$ facets (an exact formula for $$F(d,m)$$ can be found in Ziegler's "Lectures on polytopes"). By going to the dual polytope $$P^*$$ one sees that if the number $$n$$ of vertices of a $$d$$-polytope satisfies $$F(d,m) \ge n > F(d,m-1)$$, then it has at least $$m$$ facets.

For $$n = F(d,m)$$ this bound is exact (attained by the duals of neighborly polytopes). I do not know if for any $$n$$ inbetween there is a polytope with $$m$$ facets.