10 problems

Let $P$ be a Gorenstein poset, and write its $cd$-index as a homogeneous polynomial in the noncommutative variables $c$ and $d$, of degrees $1$ and $2$, respectively. Bayer–Klapper…

Finite-generation conjecture. For every positive integer $n$, this closed cone is finitely generated.

Half-Eulerian inequality-generation conjecture. Every linear form that is nonnegative on the flag vectors of all half-Eulerian posets is the sum of a linear form that is nonnegativ…

Cone equality conjecture. The two cones are equal:

Let $Q^{d,n}$ be the braxtope considered above, with $n\ge d\ge 3$. Let $P$ be the $(d-3)$-fold pyramid over the bipyramid over the $(n-d+2)$-gon. Flag-vector conjecture. The flag…

Let ${\mathcal C}$ be the cone of flag vectors of $4$-polytopes, and let $\ell_2$, $\ell_4$, and $\ell_6$ be the indicated rays of the cone ${\mathcal D}$ containing the flag-vecto…

Let $\mathcal{P}^{4}$ denote the class of flag vectors of $4$-polytopes, and let $\mathcal{S}^{3}$ denote the class of flag vectors of $3$-spheres. For a class $\mathcal{C}$, write…

Let $P_{01,d}$ be the set of all $d$-dimensional $01$-polytopes, together with their polars. A set $P$ of $d$-polytopes is broad when every word-set $bb_s$ that is nonnegative on e…

For each dimension $d$, consider the extended $g$-vector of $d$-dimensional convex polytopes and the finite calculation described in the paper. Fine's finite-calculation conjecture…

Let $X$ be a convex polytope, and let the extended $g$-vector be the vector whose components are the linear functions on flag vectors specified by the axioms in the previous sectio…