## Joshua Hinman proved Bárány’s conjecture on face numbers of polytopes, and Lei Xue proved a lower bound conjecture by Grünbaum.

### Joshua Hinman proved Bárány’s conjecture.

One of my first posts on this blog was a 2008 post Five Open Problems Regarding Convex Polytopes, now 14 years later, I can tell you about the first problem on the list to get solved.

Imre Bárány posed in the late 1990s the following question:

For a $d$-dimensional polytope $P$ and every $k$, $0 \le k \le d-1$,  is it true that $f_k(P) \ge \min (f_0(P),f_{d-1}(P))$?

Now, Joshua Hinman settled the problem! In his paper A Positive Answer to Bárány’s Question on Face Numbers of Polytopes he actually proved even stronger linear relations. The abstract of Joshua’s paper starts with the very true assertion: “Despite a full characterization of the face vectors of simple and simplicial polytopes, the face numbers of general polytopes are poorly understood.” He moved on to describe his new inequalities:

$\frac{f_k(P)}{f_0(P)} \geq \frac{1}{2}\biggl[{\lceil \frac{d}{2} \rceil \choose k} + {\lfloor \frac{d}{2} \rfloor \choose k}\biggr], \qquad \frac{f_k(P)}{f_{d-1}(P)} \geq \frac{1}{2}\biggl[{\lceil \frac{d}{2} \rceil \choose d-k-1} + {\lfloor \frac{d}{2} \rfloor \choose d-k-1}\biggr].$

### Lei Xue proved Grünbaum’s conjecture

In her 2020 paper: A Proof of Grünbaum’s Lower Bound Conjecture for general polytopes, Lei Xue proved a lower bound conjecture of Grünbaum: In 1967, Grünbaum conjectured that any d-dimensional polytope with d+s2d vertices has at least

$\phi_k(d+s,d) = {d+1 \choose k+1 }+{d \choose k+1 }-{d+1-s \choose k+1 }$

k-faces. Lei Xue proved this conjecture and also characterized the cases in which equality holds.

Congratulations to Lei Xue and to Joshua Hinman.

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### 4 Responses to Joshua Hinman proved Bárány’s conjecture on face numbers of polytopes, and Lei Xue proved a lower bound conjecture by Grünbaum.

1. Can it help shed some light on your 3^d conjecture?

2. kodlu says:

Very nice. Are you going to blog about Park and Pham’s proof of the Kahn-Kalai conjecture?

• Gil Kalai says:

Dear kodlu, I certainly should blog about the proof! (And I am also thinking about blogging about our old 2006 program toward a proof and some related open questions.)