7 problems

Let $\Delta$ be an $n$-dimensional lattice polytope whose $h^*$-polynomial is quadratic, written as … Quadratic $h^*$-polynomial conjecture. One has … This is proposed as a more pr…

Volume-boundedness conjecture. The value $h^*(1)=\operatorname{Vol}_{\mathbb N}(\Delta)$ is bounded by a constant depending only on the leading coefficient $h_d^*$ of $h^*$.

Let $\tau$ be the ordinal sum of preorders $\tau_1$ and $\tau_2$, and let $\mathcal R_\tau^\vee$ denote the polar dual of the translated reflexive preorder polytope. Ordinal-sum fa…

Let $E$ be a totally ordered $n$-element set, let $\tau$ be a preorder on $E$, and let $\operatorname{des}(w)$ be the number of descents of a word $w\in E^n$. Define…

Let $G$ be a bipartite graph, and let $P$ be a facet of its symmetric edge polytope associated with an orientation of $G$. The standard orientation is the orientation in which ever…

Let $\Delta$ be a lattice polytope whose $h^*$-polynomial is … where $b\ge 2$. Volume-bound conjecture. One has … or equivalently, … This conjecture extends the observed volume bou…

Lattice-pyramid conjecture. There is a function $f_i$ depending only on $c$ and $d$ such that any $n$-dimensional lattice polytope $P$ with $h^*_i=c$ and degree $\deg(P)=d$ is a la…