11 problems

Thin-ray conjecture. The function $f$ is unbounded below on $\mathcal P\cap\mathbb Z^n$ if and only if there exists a ray $\operatorname{ray}(y,d)$ of $\mathcal P$ such that, for e…

Let $x\in\Xi^{N\times M}$, and consider the two-dimensional integer optimal-control problem … ; equivalently, the problem is strongly NP-hard. The one-dimensional weighted analogue…

Gamma-bound dominance conjecture. For $s-q\geq 1$, we have

Let Problem be the general quadratic nonseparable resource allocation problem with generalized bound constraints, and consider its integer version, in which $x_i\in\mathbb{Z}$ for…

Let $P\subseteq\mathbb{R}^n$ be a convex body. Its lattice width is … where $\operatorname{width}_c(P)=\max_{x\in P}c^\top x-\min_{x\in P}c^\top x$. Linear flatness conjecture. The…

Let $P$ be a convex body in $\mathbb{R}^n$ with $P \cap \mathbb{Z}^n = \emptyset$, and let $f(n)$ be a function such that $w(P) \leq f(n)$, where … is the lattice width. Linear fla…

Normalizer-equivalence conjecture. If $\operatorname{Fix}(G)^{\perp}$ contains a rational $G$-invariant subspace other than $\{0\}$ and $\operatorname{Fix}(G)^{\perp}$ itself, then…

Herr–Rehn–Schürmann conjecture. If $G$ is not $2$-homogeneous, then $G$ has infinitely many core points up to translation equivalence.

Let ${\bf V}$ be the matrix appearing in a decomposition of the Gram matrix, and let $x_i$ denote the associated values. Efficient-algorithm conjecture. Particular choices of…

Let $\Gamma$ be a transitive permutation group acting on $\mathbb{R}^n$, and let a core point be an integral vector whose orbit polytope is lattice-free. Let…

Let $K\subset \mathbb{R}^n$ be a convex set, and let $w(K)$ denote its lattice width. Khinchin's flatness theorem gives a dimension-dependent function $\omega(n)$ such that…