37 problems

Let $n\ge 2$ and consider a packing of $n$ circular disks whose radii all lie in an interval $[1-\epsilon,1]$. Stability conjecture. There exists an $\epsilon>0$ such that, for eve…

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…

Torquato–Stillinger conjecture. The function $g_2({\bf r})$ is a pair correlation function of a translationally invariant disordered sphere packing in $\mathbb{R}^d$ at number dens…

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…

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

Self-dual relaxed lattice conjecture. In every dimension, the largest possible value of $r_1$ among self-dual relaxed lattices equals the smallest value of $r$ possible in Theorem…

An FM-tetrahedron is a tetrahedron formed by four pairwise disjoint spheres, with a support sphere tangent to all four and satisfying the FM-tetrahedron conditions described in the…

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…

FCC truncated-density conjecture. For all $>0$, among packings of unit balls in $^3$, the $$ -truncated density has a maximum at the corresponding FCC lattice.

Let $K\subset[1,\infty)$ be a compact set with $1\in K$. A sphere packing is $K$-admissible if all distances between distinct centers avoid $K$ in the paper's normalization, and a…

Let $L\subset\mathbb{R}^{48}$ be a lattice with minimal norm $\sqrt{6}$. Covolume conjecture. If there is $x\in L$ with $\sqrt{8}<|x|<\sqrt{10}$, then $L$ has covolume greater than…

A sphere packing in dimension $48$ is a set of congruent non-overlapping balls whose centers form a discrete subset of Euclidean space. An extremal lattice in dimension $48$ is an…

Let $T_{[abc]}$ be a lattice in the notation of the paper, and let $W$ be a frame lattice corresponding to $T_{[abc]}$. A root is a lattice vector of minimal norm as defined in the…

A graph is a generic-radii sphere-packing contact graph if it is the contact graph of a sphere packing whose radii are generic. NP-hardness conjecture. Determining whether a graph…

Let $G=(V,E)$ be the contact graph of a sphere packing $P$ with generic radii, where generic means that the radii form an algebraically independent set. Regard $P$ as a bar-and-joi…

Ball number conjecture. For any nontrivial and nonsplittable link $L$,

Consecutive-curvature sequence conjecture. There is a sequence of consecutive tangent spheres

Curvature realization conjecture. The set of curvatures of $\Omega(\mathcal{B}_{\mathcal{R}^4})$ is $\mathbb N$.

Let $d\ge 2$ and let $P$ be an edge-scribable $(d+1)$-polytope. A polytope is Möbius unique if its edge-scribed realizations are equivalent under Möbius transformations. Möbius uni…

Let $B_R$ be the radius-$R$ ball, let $\sigma_R$ be the uniform measure on $B_R$, and set $\alpha=\frac{10}{\pi}$. Define … Let $\mathcal{F}_{\sigma_R}^{\alpha}$ denote the grid-ce…

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 $\rho>0$,…

Linear-programming and sign uncertainty conjecture.

Let $\Gamma$ be a container, let $\rho(1,k\Gamma)$ denote the optimal density of unit-sphere packings in the $k$-fold magnification $k\Gamma$, and define … For a large class of con…

For $N=n+1$, let $v_1,\ldots,v_N\in\mathbb R^n$ satisfy $|v_1|=\dots=|v_N|=r$, and let $B_{v_i}(R)$ denote the ball of radius $R$ centered at $v_i$. The union-of-balls simplex conj…