12 problems

Gubeladze's conjecture. The coefficient $2$ in front of $n$ in the bound for the integer Carathéodory rank of normal polytopes cannot be improved.

Let $\operatorname{NPol}(d)$ denote the partially ordered space of $d$-dimensional normal polytopes, equipped with its order-complex structure. Emergent analytic behavior conjectur…

Let $(\operatorname{ICP})$ denote the integral Carathéodory property for normal polytopes, and consider counterexamples to this property in varying dimensions. Trivial extensions a…

Asymptotic Carathéodory-rank conjecture. The ratio $\operatorname{CR}(d)/d$ is monotonically converging to $2$ as $d\to\infty$. The source records a counterexample to the integral…

Let $P\subset\mathbb{R}^d$ be a normal lattice polytope. For every $c\in\mathbb{N}$ and every $x\in(cP)\cap\mathbb{Z}^d$, consider representations of $x$ as a weighted sum of latti…

Integral Carathéodory conjecture. Every rational finite pointed cone has the integral Carathéodory property. The polytope formulation is the corresponding special case under homoge…

Let a lattice polytope be a polytope whose vertices lie in an integer lattice. Normal polytopes are lattice polytopes for which every lattice point in $cP$ can be expressed as a su…

An integral cyclic polytope is a cyclic polytope with integral vertices. A lattice polytope is very ample when its associated semigroup is generated up to saturation in the usual t…

Let $C_d(\tau_1,\dotsc,\tau_n)$ and $C_d(\tau_1',\dotsc,\tau_n')$ be integral cyclic polytopes, and let $\Delta_{ij}$ denote the corresponding gaps used in the source. Assume that…

Let $C_4(\tau_1,\tau_2,\dotsc,\tau_n)$ be a cyclic polytope, and write $\Delta_{ij}=\tau_j-\tau_i$ for $i<j$. A lattice polytope is normal when every lattice point in every positiv…

A cyclic polytope is the convex hull of points on the moment curve; here, normal means that every lattice point in every positive dilate is a sum of lattice points of the polytope.…

A simple lattice polytope is a lattice polytope in which exactly $\operatorname{dim} P$ edges meet at every vertex. An edge is called long when its lattice length satisfies a unifo…