110 problems

Let $P$ be a $d$-polytope with vertex set $V$, and let $d'\le d-1$. Suppose that for every integer $k$ with $0\le k\le d'$, condition (dfcon) is satisfied. Write $i(P,m)$ for the E…

Dickenstein–Nill's Cayley decomposition conjecture. If

Let $\Delta$ be a smooth fibre-like polytope of dimension $d$, and let $X(\Delta)$ denote the associated smooth toric Fano variety. Assume that $d$ is an odd prime number. Classifi…

Let $\Delta$ be a Gorenstein polytope of CY-dimension $n$, so that its stringy E-function $E_{\rm st}(\Delta;u,v)$ and CY-dimension are defined as in the preceding discussion. Stri…

Let $T$ be a lattice tetrahedron with exactly $k$ interior lattice points. In a minimal direction for its lattice width, consider the consecutive lattice planes meeting $T$. Lattic…

Volume lower-bound conjecture. Every $P\in\mathcal{P}_o^n$ satisfies

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…

Let $p,q \in \mathbb{Z}$ be coprime integers with $1<q<p^2$, with $p$ odd and $q$ even. Let $\Delta(p,q)$ denote the triangle defined in the paper, and suppose that Pick's formula…

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…

Real-part bound. All roots $\alpha$ of Ehrhart polynomials of lattice $d$-polytopes satisfy

Let $M$ be a lattice and let $M_{\mathbb{R}}=M\otimes_{\mathbb{Z}}\mathbb{R}$. For a lattice polytope $P\subseteq M_{\mathbb{R}}$, let $L(P)$ denote its lattice length, and write…

Boundary-vertex conjecture. The vertices of $Q$ lie on the boundary of $K-\mathbf{x}$.

Let $P$ be the base polytope of a polymatroid. A triangulation of $P$ is unimodular if all its simplices are unimodular with respect to the relevant lattice, and it is flag if ever…

Let $E$ be an $n$-element set and let $\tau$ be an arbor on $E$. Its $h$-polynomial is $h(\tau,t)=\sum_{i=0}^n h_i(\tau)t^i$, where $h_i(\tau)$ counts the lattice points of the arb…

Width-one conjecture. Under the stated hypothesis, $P$ has width one.

Let $\omega=(\omega_0,\dots,\omega_d)\in\mathbb{R}^{d+1}_{>0}$ satisfy $\omega_0\leqslant\dots\leqslant\omega_d$, and define the weighted terminal simplex … For $i\in[d]$, Codenott…

Let $T_d=\operatorname{conv}(-\mathbb{1}_d,e_1,\dots,e_d)$ be the terminal simplex in dimension $d$, and let $i\in[d]$. Merino–Schymura conjecture. For every $d\in\mathbb{N}$ and e…

Let $P\subseteq \mathbb{R}^d$ be a non-hollow lattice polytope. A full-dimensional lattice polytope is called terminal if it is a direct sum of translates of terminal simplices, wh…

Let ${\mathcal P} \subset {\mathbb R}^n$ be a normal polytope of dimension $n$ with $({\mathcal P} \setminus \partial {\mathcal P}) \cap {\mathbb Z}^n \neq \emptyset$. For…

Oda's conjectures. The following two statements are true: (1) every smooth polytope has the IDP; and (2) if $P,Q\subset\mathbb{R}^n$ are smooth polytopes and the normal fan of $P$…

Let $P \subset N_{\operatorname{\mathbb{Q}}}$ be a smooth Fano polytope of dimension $n$. A smooth Fano polytope is a lattice polytope whose vertices are primitive lattice points a…

Let $S \subset \mathbb{Z}^d$ be a bounded set, and let $\beta_{\mathbb{Z}}(S)$ denote its discrete Borsuk partition number. Let $\operatorname{conv}(S)$ be the convex hull of $S$,…

Let $P$ be a lattice $d$-polytope, and let $\ell \in [d]$. Define $\operatorname{LD}_P^{\ell}(k)$ to be the number of $\ell$-dimensional slices of $P$ that contain the most lattice…

Let $d \geq 3$ and let $P$ be a lattice $d$-polytope. Lattice-polytope diameter conjecture. Computing a lattice diameter of $P$ is an $\operatorname{\mathsf{NP}}$-hard problem. Thi…

Let $P$ be an IDP polytope, meaning a lattice polytope such that for every $k\in\mathbb{N}_{\ge 2}$, each lattice point in $kP$ is a sum of $k$ lattice points in $P$. If … is the…