103 problems

Voronoi's conjecture. Every parallelohedron is an affine transformation of a Dirichlet–Voronoi cell of some lattice.

Betke-Henk-Wills conjecture. For any $K\in\mathcal{K}^d_0$ and $\Lambda\in\mathcal{L}^d$,

Let $K\subseteq\mathbb{R}^d$ be a convex body centred at the origin, meaning that its centre of mass is the origin. Assume that the interior of $K$ contains no lattice point other…

Let $X_d$ be the space of lattices, let $\mathcal{WR}_d$ denote the set of well-rounded lattices, and consider the covering-radius condition from the source: for every…

Let $L\subset {\mathbb R}^n$ be a unimodular lattice, and define the multiplicative norm of ${\boldsymbol x}=(x_1,\ldots,x_n)$ by $N({\boldsymbol x})=|x_1\cdots x_n|$. Minkowski's…

Let $X_d$ be the space of lattices under consideration, let $A$ be the group acting on $X_d$, and let $\mu=\operatorname{CovRad}_N$ be the covering radius associated with…

Let $\Lambda\subset\mathbb{R}^d$ be a $d$-dimensional lattice. For a convex body $K\subset\mathbb{R}^d$, define its lattice-normalized volume by … Consider the maximum of…

Three-dimensional flatness conjecture. The three-dimensional flatness constant satisfies

Minkowski-type criterion for three-dimensional lattice coverings. The arrangement $K+\mathbb{Z}^3$ is a lattice covering of $\mathbb{R}^3$ if and only if there exist a unimodular t…

Covering product conjecture. One has

Let ${\mathcal{L}_n}$ denote the space of unimodular lattices in ${\mathbb{R}}^n$, and let $\kappa_n$ be the infimum of their Mordell constants, where the Mordell constant of a lat…

Let $B_n$ be the closed Euclidean unit ball in ${\bf R}^n$, let $K_n$ be the unit cube in ${\bf R}^n$, and let $\beta(B_n,K_n)$ denote the corresponding lattice covering quantity.…

Euclidean norm conjecture. The Euclidean norm has this property: the sequence of all $f$-best simultaneous approximations can have any given sequence of signatures.

A two-dimensional (or, more generally, $n$-dimensional) continued fraction is represented by a sail, whose facets have homeomorphic types, adjacency relations, integer volumes and…

Let $L=C\mathbb{Z}^n\subset\mathbb{R}^n$ be a lattice with Gram matrix $G=C^{\mathsf T}C$ and associated flat torus $T_L=\mathbb{R}^n/L$. Normalize to unit volume, so $\det G=1$. I…

Let $G=\mathbb{Z}/p\mathbb{Z}$ with $p$ an odd prime, let $\Bbbk$ be a field of characteristic different from $p$, and let $V$ be a representation of $G$. Let $m$ be the number of…

Let $d\ge 3$, and let $w(t)$ be a decreasing real function tending to zero as $t\to+\infty$ and satisfying … The theorem asserts that if the series … converges, then for every posi…

Let $K$ be a convex body in $\mathbb{R}^{n}$, and let $(K-K)^{\circ}$ be the polar body of its difference body. Makai Jr.'s conjecture. … Equality should hold if and only if $K$ is…

Let $n\geq 2$ and let $K\subset\mathbb R^n$ be an origin-symmetric convex body satisfying … where $c>0$ is a universal constant. Convex-body extension conjecture. There exists a la…

Let ${\boldsymbol{B}}_{14}(a)$ be the parameterized $14$-dimensional lattice generator, and let $f(v)$ be the polynomial defined in the paper. Let $v_0$ be the second positive root…

Let ${\boldsymbol{B}}_{13}(a_1,a_2,a_3)$ be the parameterized $13$-dimensional lattice generator, and let $f^B_2$ and $f^B_3$ be the two degree-$14$ polynomials defined in the pape…

Let ${\boldsymbol{B}}_{13}({\boldsymbol{a}})$ be the three-parameter family of $13$-dimensional lattice generators described in the paper, with ${\boldsymbol{a}}=[a_1,a_2,a_3]$. Op…

Let $a_{\text{opt}}$ be the parameter at which the necessary local-optimality condition $G'(a)=0$ holds, and let ${\boldsymbol{B}}_{14}(a)$ denote the parameterized $14$-dimensiona…

Let a finite collection of lattice vectors spanning $\mathbb{R}^d$ be cosimple if it has a linear dependence whose coefficients are all non-zero and have pairwise different absolut…

Let $Z=\sum_{i=1}^n[\mathbf{0},\mathbf{u}_i]\subseteq\mathbb{R}^{n-1}$ be a lattice zonotope of dimension $n-1$ with $n$ generators, whose every $n-1$ generators are linearly indep…