33 problems

Let $0<\mu<1$, and let $c_i$ be circles of radii $r_i$. Associate to each $c_i$ a concentric circle of radius $\mu r_i$, called its kernel. A generalized Minkowskian arrangement of…

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…

Fejes Toth's conjecture. The packing in Figure has maximum density among disks with two radii whose ratio is slightly greater than $\sqrt{2}-1$.

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…

Layered extremal word conjecture. If $\Pi$ is a set of layered patterns, then

Let $\triangle_n(r)$ denote the supremum of the densities of nice $r$-sphere packings in hyperbolic $n$-space. A packing is periodic if its density arises from a periodic packing.…

Let $a<b$ be coprime positive integers, and let … For a finite set $S\subset\mathbb{Z}$, write $d_p(S)$ for its packing density and $d_c(S)$ for its covering density. The packing a…

Connelly's uniformity conjecture. The ternary triangulated packing numbered $53$ is the most uniform triangulated packing other than HCP. Its proposed deformation yields the highes…

Density-maximizing conjecture. For the ratios $r_b$ and $r_c$, the periodic nontriangulated binary packing depicted in Fig. 7 maximizes density among all binary packings with the s…

A saturated compact packing of disks with $n$ different radii is a compact packing in which at least one disk of each size appears and no further disk of the smallest relevant radi…

For a body $K$ in $E^n$, let $\Delta_k(K)$ be the infimum of the densities of all $k$-saturated packings by congruent copies of $K$, where deleting $k-1$ members never creates a vo…

For a centrally symmetric convex disk, define packing closeness and covering looseness using the largest negatively homothetic copy of the disk. Linhart's conjecture. Every central…

Molnár's strip-packing conjecture. The maximum density is

An $r$-accessible packing is a packing in which a cellarer whose vertical shadow on the floor is a circle of radius $r$ can freely come into contact with every barrel. Heppes's con…

Let a blocking-free packing of the plane by congruent circles be a packing in which every circle can be moved arbitrarily far from its original position through the uncovered part…

Kertész's density conjecture. If ${\bf Q}$ contains a TS-packing of $N$ unit balls, then

Let $B^d$ be the unit ball in $d$-dimensional Euclidean space, let $\delta_\rho(B^d,n)$ be the optimal parametric density for a finite packing of $n$ balls with parameter $\rho>0$,…

Consider packings by $n$ different given discs. A packing is saturated if no further disc can be added. Saturation conjecture. If the densest compact packing is saturated, then the…

For a $d$-dimensional convex body $C$, let $\delta(C)$ and $\theta(C)$ denote its packing and covering densities by congruent copies. Dimension-independent packing–covering conject…

Let $d\geqslant 2$ be fixed. For a $d$-dimensional convex body $C$, let $\delta(C)$ and $\theta(C)$ denote its packing and covering densities by congruent copies. Kuperberg's conje…

Binary sphere packing conjecture. The most dense packing has density $0.793$, achieved when the large spheres are centered at the face-centered cubic lattice.

Isostatic conjecture. The density $\rho$ of every such packing satisfies

Uche–Stillinger–Torquato conjecture. Some of the packings in the tricusp case are most dense.

Square-torus Markov-bound conjecture. Every such packing satisfies $M\leq 6.25$. Equivalently,

Let $N$ be the number of equal disks in a square torus, let $\delta(N)$ denote the packing density, and let $\pi/\sqrt{12}$ be the density of the planar hexagonal packing. Square-t…