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.