121 problems

The case of unit spheres in three-dimensional Euclidean space concerns packings of congruent spheres, whose density is the proportion of space they occupy. Kepler's conjecture. The…

In the space $X^n$, let $d_n(r)$, for $n\geq 2$, denote the density of $n+1$ mutually tangent spheres, or horospheres when $r=\infty$, of radius $r$ relative to the simplex spanned…

Let $\mathbb S^{n-1}=\{x\in\mathbb R^n:\|x\|=1\}$ be the unit sphere, and let $\alpha_n$ be the supremum of the normalized surface measure of a measurable subset of…

Let $N>1$ and $3\leq d\leq 41$. Consider an arbitrary TS-packing of $N$ unit balls in $\mathbb{E}^d$. Fejes Tóth's sausage conjecture. The volume of their convex hull is at least t…

Contact-number equality conjecture.

Local-global conjecture for Apollonian bends. All sufficiently large integers satisfying certain congruence restrictions appear as the bends of some circle in the Apollonian circle…

A sphere packing is a collection of congruent non-overlapping spheres in Euclidean space; it is disordered when it lacks the long-range order characteristic of a lattice packing. T…

Consider packings of unit balls in $\mathbb{R}^4$, and write $\Delta_4$ for the maximal packing density, $B_4$ for the volume of a unit ball, and $\delta_4=\Delta_4/B_4$ for the ma…

A packing of congruent spherical caps can be studied by decomposing the sphere into simplices whose vertices are cap centers; the density in each simplex is compared with the densi…

Let $P$ be any packing of unit spheres in $\mathbb{R}^4$, and consider its Voronoi decomposition. Let the volume of a regular 24-cell circumscribed about a unit sphere be the refer…

A lattice $L\subset\mathbb{R}^{48}$ determines a lattice sphere packing by placing congruent balls at its lattice points; its density is maximized when the ratio of the ball radius…

Möbius uniqueness conjecture. Every edge-scribable $(d+1)$-polytope is Möbius unique.

Let $\Lambda$ be a lattice in three-dimensional Euclidean space, and for $s>3$ define its Epstein zeta function by … The FCC lattice denotes the face-centered cubic lattice. Sarnak…

Consider a packing of unit balls in three-dimensional Euclidean space, and let a Voronoi cell be the region of points at least as close to its ball's center as to the centers of al…

Let $n$ satisfy $2\leq n\leq 8$, and let $2^k$ be the largest power of $2$ strictly less than $n$. A sphere packing is weakly recurrent and dense when it has the corresponding recu…

Let $D^*$ be a Delaunay star in a saturated packing, and let its score be the sum of the scores of its standard clusters. The Delaunay stars of the face-centered cubic and hexagona…

Let a Delaunay star be the finite configuration associated with a vertex in the Delaunay decomposition of a saturated packing, and let its score be the sum of the scores of its sta…

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…

A stable representation is an ${\mathcal{M}}$-representation that is a local minimum with respect to the ordering relation defining stability. Consider a graph on the sphere and fi…

Assume that all nonzero vectors of a lattice $\Lambda\subset\mathbb{R}^{24}$ have length exactly $2$ or at least $\sqrt{6}$. Let $g\colon\mathbb{R}^{24}\to\mathbb{R}$ be a Schwartz…

A lattice $\Lambda\subset\mathbb{R}^n$ is a discrete subgroup of rank $n$, and its packing density is the density of the packing by spheres centered at lattice points with radius h…

Maximum-score conjecture. The maximum of $\sigma$ on $X$ is the constant