27 problems

Let $P$ be a reflexive polytope, meaning a lattice polytope whose polar dual is also a lattice polytope. Let $h^*_P(x)$ denote its $h^*$-polynomial. Hibi's conjecture. Every reflex…

Let $P$ be a reflexive polytope of dimension $d$, and write $\lvert V(P)\rvert$ for its number of vertices. Nill's conjecture. The number of vertices satisfies … This is a proposed…

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…

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…

Golyshev's canonical line hypothesis. Every $d$-dimensional smooth Fano polytope with $d\leq 5$ satisfies the canonical line hypothesis. This connects the roots of Ehrhart polynomi…

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…

Let $\tau$ be a preorder, let $\mathcal R_\tau$ be the translated reflexive preorder polytope, and let $\mathcal R_\tau^\vee$ be its polar dual. Polar h-star duality conjecture. ……

Nill's conjecture.

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…

Let $\mathcal{X}$ be a toric Calabi–Yau $n$-fold whose toric diagram is a reflexive $(n-1)$-dimensional polytope. Let $V_{\min}$ be its minimized volume, let $\chi$ denote the Eule…

Let $Q\subset\mathbb{R}^{d}$ be a $d$-dimensional $\ell$-reflexive polytope with dual $Q^{\ast}=\ell Q^{\circ}$, where $d\geq4$. Let $\mathcal{F}_{d-2}(Q)$ and…

Let $Q esubsetmathbb{R}^{3}$ be a $3$-dimensional $ell$-reflexive polytope with dual $Q^{ast}=ell Q^{circ}$. Let $Lambda{operatorname{Edg}(Q)}$ and…

Nonexistence conjecture. There do not exist reflexive polytopes satisfying $(**)$ for the pairs $(Z_{13},J_{3,0})$, $(Z_{17},Q_{2,0})$, $(U_{16},U_{16})$, $(W_{17},S_{1,0})$,…

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

Half-strip conjecture. Every root $z$ of $L_P$ satisfies

CL-polytope lattice-point conjecture. If $P$ is a $d$-dimensional CL-polytope, then

Batyrev–Borisov vertex-maximality conjecture. The polytope $P_6^{\oplus d/2}$ has the maximal number of vertices among reflexive polytopes in even dimension.

Let $n\geq 4$ and let $X$ be a canonical Gorenstein Fano variety of dimension $n$. Let $t_{n-1}$ and $y_0,\ldots,y_{n-2}$ denote the entries of the Sylvester weight system of lengt…

Let $T_{7}$ be the polyhedral triangulation discussed above, and call a polyhedral triangulation reflexive when its associated cone is reflexive. Uniqueness conjecture for $T_{7}$.…