8 problems

Let $K$ be a convex disk, and let $\vartheta^1_T(K)$ and $\vartheta^1_L(K)$ denote its $1$-fold translative and lattice covering densities, respectively. Bambah–Rogers conjecture.…

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…

Let $p$ range over $1<p<\infty$, and consider the curve whose value at $p$ is the maximum covering constant for the Minkowski ball $D_p$. The curve of maxima conjecture. The curve…

Let $T$ be a regular tetrahedron in three-dimensional space, and let $\vartheta_L(T)$ denote its lattice covering density. Forcade–Lamoreaux conjecture. … The value is supported by…

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…

A covering of the plane by unit circles has margin $\mu\in[0,1]$ if removing any one circle creates empty space that can be covered by a circle of radius $1-\mu$. Bezdek–Kuperberg…

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…