24 problems

Bezdek's conjecture.

For the one-sided kissing number $B(n)$, let $k(n)$ be the kissing number in $n$ dimensions and define … The lattice constructions in dimensions $5$, $8$, and $24$ give the lower b…

Let $P$ be a maximal kissing arrangement of points on the unit $3$-sphere with pairwise angular separation at least $\pi/3$, and let $(a,b)$ denote the parameters used in the paper…

Let $H$subset ${\bf R}^4$ be a closed half-space, and let $S$ be a unit sphere in $H$ tangent to the supporting hyperplane of $H$. The one-sided kissing number $B(4)$ is the maximu…

Let $S_i$ be a subset of shortest Leech lattice vectors whose pairwise cosine values are bounded by $1/4$, with $|S_i|=496$, and let the associated construction define a 25-dimensi…

Ball density–kissing-number conjecture. One may conjecture that

Cross-polytope kissing number conjecture. In $\mathbb{E}^4$,

Cross-polytope lattice density conjecture. In $\mathbb{E}^4$,

Let $b\mathcal{C}_nd$ be an infinite family of linear codes with $b\mathcal{C}_n\subseteq\mathbb{F}_2^n$. For a code $b\mathcal{C}$, let $b\kappa_0(\mathcal{C})$ denote the number…

Let $B^n$ denote the unit ball in Euclidean space $\mathbb{E}^n$, and let $\kappa^*_{\alpha}(B^n)$ be the generalized lattice kissing number at parameter $\alpha$. Conjecture 8.1.…

Let $B^n$ denote the unit ball in Euclidean space $\mathbb{E}^n$, and let $\kappa^*_{\alpha}(B^n)$ be the generalized lattice kissing number at parameter $\alpha$. For the Leech la…

For real $c>1$, consider the Minkowski spheres … The kissing number is the maximum number of non-overlapping congruent spheres tangent to a given sphere. Conjecture on Minkowski-sp…

Let $S^3$ be the unit sphere in $\mathbb{R}^4$. A spherical code is a finite subset of $S^3$, and it is universally optimal if it minimizes the energy … for every absolutely monoto…

For a bounded set $K\subset\mathbb{R}$, let … Let $\mathrm{kiss}_d$ be the kissing number in $\mathbb{R}^d$. Kissing-number conjecture. For every dimension $d$, … The evident lower…

Let a family of congruent disks in the plane be arranged so that no two disks intersect in their interiors, with one central disk, a first family of disks touching it, and a second…

Consider a packing of congruent balls in Euclidean space in which every ball is touched by exactly twelve other balls. Fejes Tóth's conjecture. The packing must consist of parallel…

Let $B^2$ denote the unit disk, and let $N_k(D)$ be the maximum total number of the $j$-th neighbors of a convex disk $D$ for $j=1,\ldots,k$, where neighbors are defined recursivel…

Let $N(S)$ be the maximum number of neighbors that one member can have in a packing of congruent copies of a convex disk $S$, and let $n$ be a positive integer. Conjecture on regul…

Zong's conjecture. There exists an absolute constant $c>1$ such that

For $d\geq 1$ and angular diameter $\theta$, let $k_S(d,\theta)$ be the maximum number of disjoint spherical caps of angular diameter $\theta$ in ${\mathbb S}^d$ that can be arrang…

Asymptotic one-sided kissing-number conjecture.

Let $B(n)$ denote the maximum number of unit non-overlapping spheres in a closed half-space of ${\mathbb R}^n$ that can touch a unit sphere tangent to the bounding hyperplane. Musi…

Let $\mathcal M_g$ denote the moduli space of closed hyperbolic surfaces of genus $g$, and let $\operatorname{Kiss}(S)$ be the number of systoles of $S$. Schmutz Schaller's kissing…

A kissing configuration in $\mathbb{R}^n$ is an optimal arrangement of points on the unit sphere whose pairwise inner products are at most $1/2$; a contact is a pair of points with…