Home Online Mathematical Communication Combinatorics and more 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 on face numbers of polytopes, and Lei Xue proved a lower bound conjecture by Grünbaum.

Combinatorics and more 2022-04-29

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+s≤2d 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.