184 problems

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 $P$ 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…

Let $P\subset\mathbb{R}^d$ be a lattice polytope. It has the integer decomposition property (IDP) if every lattice point of $\operatorname{cone}(P)$ is a nonnegative integral combi…

Ohsugi–Tsuchiya conjecture. The Ehrhart $h^*$-polynomials of symmetric edge polytopes are $\gamma$-positive.

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 $M$ be a matroid and let $P_M$ denote its base polytope. A triangulation of $P_M$ is unimodular if every simplex in the triangulation is unimodular with respect to the lattice…

Let $P$ be a lattice polytope in $\mathbb{R}^d$, and let $h^*_{rP}(z)$ denote its unweighted Ehrhart $h^*$-polynomial after dilation by the integer $r$. Beck–Stapledon conjecture.…

Recall that a rational polytope is smooth if it is simple and every normal cone is unimodular, and reflexive if it is a lattice polytope with the origin in its interior and its pol…

Let $P\subset\mathbb{R}^d$ be a lattice polytope of dimension $d$ and degree $s$. A Cayley polytope is a lattice polytope expressible as a Cayley sum of lattice polytopes. Cayley c…

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 $P\subset N_\mathbb{R}$ be an $n$-dimensional polytope whose dual polytope $P^*$ is a smooth projective lattice polytope in $M_\mathbb{R}$. Generalized Ewald conjecture. Then…

Let $G$ be a finite group, let $M$ be a lattice of rank $d$, and let $\rho:G\to\operatorname{Aff}(M)$ be an affine representation. Let $P\subset M_\mathbb{R}$ be a $G$-invariant…

Neatness conjecture. All smooth lattice polytopes are neat.

Haase–Ziegler's conjecture. These are all the empty $4$-simplices of width larger than two.

Let $S$ be a clean three-dimensional lattice simplex, meaning that the only lattice points on its boundary are its vertices, and suppose that $S$ has $k\geq 1$ interior lattice poi…

Kirillov's unimodality conjecture. For any $n\geq 1$, the Ehrhart $\delta$-polynomial $\delta(\mathcal{P}_n;t)$ is unimodal.

Let $P$ and $Q$ be lattice polytopes in $\mathbb R^n$. For lattice polytopes, write … A lattice polytope is smooth if it is simple and the primitive facet normal vectors around eve…

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…

Duong–author sharpness conjecture. These bounds are sharp for all $k\ge 1$: the lower equality case is represented by $T_{3k,3k,3k+1}$, and the upper equality case by…

Let $T$ be a clean lattice tetrahedron, and let $i(T)$ denote its number of interior lattice points. Its lattice width is the minimum, over nonzero integer linear forms, of the num…

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…

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 $p,q \in \mathbb{Z}$ 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…