25 problems

Let $G$ be a graph of genus $g\geq 1$, let $L_G$ denote the associated lattice, and let $\triangle$ be the standard simplex. For positive reals $\alpha,\bar{\alpha}$, let…

Algebraic intersection criterion. The arrangement $\lambda K+\mathbb{Z}^3$ is a lattice covering of $\mathbb{R}^3$ if and only if there exists a unimodular transformation…

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 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 $Z_{\mathbf v}$ be a Strong Lonely Runner Zonotope, meaning a Lonely Runner Zonotope associated with a velocity vector $\mathbf v=(v_1,\dotsc,v_n)\in\mathbb{Z}_{>0}^n$ whose en…

Let $q$ be a prime power and let $\operatorname{PRS}(k)$ denote the projective Reed–Solomon code of dimension $k$ and length $q+1$. For $2\le k\le q-2$, let…

Let $\Lambda\subset\mathbb{R}^D$ be a lattice with covering radius $\rho$, the smallest radius such that the collection of radius-$\rho$ balls centered at points of $\Lambda$ cover…

Let $s_{2n}$ denote the best covering radius of a configuration of $2n$ points on $S^{n-1}$, and let $s_{2,2n}$ and $s_{3,2n}$ denote the corresponding covering-radius quantities a…

Average-distance conjecture. For Euclidean distances, the average distance from a uniformly distributed point in $\mathbb{R}^D$ to the nearest point of $\mathcal{L}$ is not less th…

Covering conjecture. The covering radius satisfies

Let $G$ be an undirected, connected multigraph of genus $g\geq 1$. Let $H_0$ be the degree-zero hyperplane, let $P_1$ be the unit ball of the $\ell_1$-norm in $H_0$, and let…

Let $G$ be an undirected, connected multigraph with genus $g\geq 1$ and $n\geq 2$ vertices. Let $\mathcal{N}_G$, ${\rm Crit}_{\triangle}(L_G)$ and…

Let $S=\operatorname{conv}(\{v_0,\ldots,v_d\})$ be a $d$-simplex with $\mathbf{0}\in S\setminus\{v_0,\ldots,v_d\}$ and with rational vertex directions. For each $i$, let…

Let $k,d\in\mathbb{N}$ be nonnegative integers with $k\geq2$. For every lattice $d$-polytope $P$ with $k$ interior lattice points, the prescribed-interior-points conjecture. … with…

Let $S$ be a $d$-simplex with the origin in its interior and with rational vertex directions. Let $\operatorname{Vol}_{\mathbb{Z}^d}(S)$ be normalized volume and let … where…

Let $P \subseteq \mathbb{R}^d$ be a non-hollow lattice $d$-polytope. The maximum covering radius conjecture. … with equality if and only if $P$ is obtained by direct sums and/or tr…

Hou's conjecture. The covering radius of $RM(2,7)$ is $40$.

Stable-lattice covering-radius conjecture. One has

Let $\ell_q(r,3)$ be the length function, let $\ell_q(r,3,5)$ be the $d$-length function, let $s_q(r-1,2)$ be the smallest size of a $2$-saturating set in $\mathrm{PG}(r-1,q)$, and…

Let $S=\{\boldsymbol{z}_1,\dotsc,\boldsymbol{z}_m\}\subseteq\mathbb{Z}^n$ and let $Z$ be the lattice zonotope generated by $S$. The set $S$ is in LGP (general linear position) when…

The $\ell_2$ Kannan–Lovász conjecture. The covering radius $\mu(\mathcal{L})$ should be at most $M(\mathcal{L})$ up to polylogarithmic factors in $n$.

Let $mathcal E(n)$ be the projected lattice points on $S^2$, and let $N_n$ denote their number. The covering radius $M(P_1,\dots,P_N)$ of a configuration on $S^2$ is the least…

Let $X_N$ consist of $N$ points chosen independently and uniformly with respect to surface measure on the unit sphere $\mathbb{S}^d$ in $\mathbb{R}^{d+1}$. Let…

Let $Q\subset \mathbb F_2^n$ be an $\mathbb F_2$-linear code of size $n^t$, where $t=\Theta(1)$. Its covering radius is the maximum, over $u\in\mathbb F_2^n$, of the minimum Hammin…

Let $Q\subset \mathbb F_2^n$ be an $\mathbb F_2$-linear code, and let $\mu_{Q+u}$ be the uniform probability distribution on the coset $Q+u$. Write $\overline{\mu_{Q+u}}$ for its w…