13 problems

Let $K$ be a convex body in $\mathbb{E}^3$. The density of a packing is the proportion of space occupied by the translates of $K$, and the densest packing density of $K$ is the sup…

Let $f(n)$ be the maximum possible sum of the side lengths of $n$ open, non-overlapping squares contained in a unit square. Campbell–Staton conjecture. For any integer $c$ and any…

Let $\gamma(\mathbb{R}^n)$ and $\gamma^*(\mathbb{R}^n)$ denote the two packing constants for Euclidean $n$-space, and let $\delta_n$ and $\delta_n^*$ denote the corresponding Eucli…

Rigid-transformation packing conjecture. Under this dimension hypothesis, the set obtained by applying the transformations in $\Theta$ to $E$ has positive $d$-dimensional Lebesgue…

For each integer $m\geq 2$, let $(Y_m,\omega_m)$ be the toric symplectic manifold obtained from $\mathbb{CP}^{2}$ with lines of area $m+2$ by symplectically blowing up along the st…

Let $L_{1,1}$ denote the Clifford torus in $(\mathbb{CP}^{2},\omega_{\mathrm{FS}})$, and let an integral packing mean a packing by disjoint monotone Lagrangian tori whose associate…

Let $L_{3,6}$ and $L_{2,7}$ be the indicated $L$ polyominoes, and let $cp^{free}(L_{a,b})$ denote their clumsy free packing numbers. The preceding constructions give packing number…

Let $D$ be a finite set of positive integers. A $D$-avoiding set is a set $S$ of nonnegative integers containing no two elements whose difference lies in $D$, and a winner for $D$…

Planar equilateral-triangle packing conjecture. The value of $\mathcal{N}_2\Big(B_2(0,R+\frac{r}{2}),\frac{r}{2}\Big)$ can be obtained by taking the extremities of non-overlapping…

Let $\delta$ denote the fraction of empty space in a packing of regular tetrahedra, so that the packing density is $1-\delta$. Gravel–Elser–Kallus conjecture. The optimal packing d…

Let $C\approx 0.74759$ denote the one-dimensional packing density for the saturated car parking model, and consider the analogous model in $n$-dimensional space with cars given by…

Superball lattice-packing conjecture. The densest packings of superballs for all convex shapes, $1/2\leq p\leq\infty$, are certain lattice packings, depending on the deformation pa…

Non-lattice-packing conjecture. The optimal packing of any convex, congruent polyhedron without central symmetry generally is not a lattice packing.