14 problems

Let $d\geq 4$, and let $S_{Q'_d}$ be the reflexive simplex defined from the Sylvester-sequence weight system in the source. A $d$-dimensional lattice polytope is assumed to have ex…

Simplicial vertex bound conjecture.

Vertex bound conjecture.

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…

Nill's conjecture.

Let $P$ be a reflexive polytope. Let $\Sigma$ be a simplicial fan whose ray primitive generators are the vertices of $P$, and let $\mathcal{V}$ be the fan of the canonical bundle o…

Let $X(\Delta_{n-1})$ be the toric variety associated with a reflexive toric diagram $\Delta_{n-1}$, and let $\widetilde{X(\Delta_{n-1})}$ denote the relevant resolution. Let…

Let ${\mathcal P}$ be a $d$-dimensional reflexive polytope, meaning a lattice polytope whose polar dual is also a lattice polytope after placing the origin in its interior. Let…

Let $\Delta$ be a reflexive simplex, and let $\mathrm{mldeg}(\Delta)$ and $\mathrm{deg}(\Delta)$ denote its maximum likelihood degree and degree, respectively, under the standard s…

A five-dimensional reflexive polytope is a lattice polytope whose only interior lattice point is the origin and whose polar dual is also a lattice polytope. Such a polytope gives r…

Let $(P_j)_{j=0}^{\infty}$ be a family of reflexive polytopes whose Ehrhart polynomials $f_j(x)=H_{P_j}(x)$ form an orthogonal polynomial sequence and satisfy … with…

Nonreal-root real-part conjecture. Then

Real reflexive-polytope volume conjecture. If $P$ is $d$-dimensional, then

Chan–Leung's facet-point conjecture. If $b_P=0$, then for every facet $F$ there exists a point $x_F\in\operatorname{aff}(F)$ such that