### 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 -dimensional polytope and every , , is it true that ?

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:

### 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+s≤2d* vertices has at least

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

**Congratulations to Lei Xue and to Joshua Hinman.**

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

I dont know.

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

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.)