One of Barany's most intriguing conjectures is about the $f$-vectors of convex polytopes. It asks:

Let $P$ be a convex $d$-polytope. Is it always true that $f_k \geq \min(f_0, f_{d-1})$?

A ** convex $d$-polytope** is the convex hull of finitely many vertices $v_1, \ldots, v_n \in \mathbb{R}^d$. A

**of the polytope is whatever you can shave off with a hyperplane. Its dimension is the dimension of the affine hull it generates. The number of faces are collected in the so-called $f$-vector $f = (f_0, \ldots, f_{d-1})$, where $f_k$ denotes the number of $k$-dimensional faces. Further information on polytopes can be found in Ziegler's "Lectures on Polytopes".**

*face*My question is

What's the status of Barany's conjecture? For which classes of polytopes is it known to hold?

Here is what I know:

By double counting it is easy to see that the conjecture is true for simple and simplicial polytopes. Furthermore, it is nothing but a boring calculation to prove that the basic polytope constructions like taking products, prisms, pyramids, etc. retain the the conjectured inequality.

Given that the face lattice of a convex $d$-polytope is a graded, atomic, coatomic lattice of rank $d+2$, it makes sense to ask a far more general question: Given such a lattice, denote the number of elements of the $k$th level again by $f_k$. Is it still true that $f_k \geq \min(f_0, f_{d-1})$? Here the answer is no since I was able to construct a counterexample. But if the lattice also has what Ziegler calls the "diamond property" (every interval of length two looks like a diamond), then I don't know the answer.