12 problems

Pach's decomposition conjecture. For every convex disk $K$, there exists a minimal natural number $m(K)$ such that every $m(K)$-fold covering of the plane by translates of $K$ can…

Let $\mathcal C$ be a family of unit disks forming an $m$-fold covering of the plane, meaning that every point belongs to at least $m$ members of $\mathcal C$. Winkler's conjecture…

Hadwiger's covering-number conjecture. One should have

Complete saturation and reduction conjecture. Every body $K$ in $\mathbb{E}^d$ (respectively, in $\mathbb{H}^d$) admits a completely saturated packing and a completely reduced cove…

Let $d$ be a degree and let $\vec{e}=(e_1,\ldots,e_{d-1})$ be a $(d-1)$-tuple of scrollar invariants. A cover is primitive if it does not factor through a nontrivial proper subcove…

Let $\mathcal C$ be a sufficiently thick covering of the plane by unit disks or by smooth convex objects, meaning a covering with enough multiplicity that its members might be deco…

Let $p$ be the parameter of the family of Minkowski balls and let the covering constant be the associated covering invariant. Conjecture on the curve of maxima. The curve of maxima…

In hyperbolic three-space $\mathbb{H}^3$, consider the horoball covering belonging to the $\{6,3,3\}$ tiling, whose density is approximately $1.280$. Fejes Tóth's conjecture. This…

Let $\pi_2$ be the elliptic fibration constructed from a point of ${\mathcal M}'_{K,1}$, let $\sigma_0$ be its zero section, and let $\phi$ be the map associated with the correspon…

Let ${\mathcal L}$ be a rigid line configuration, and let $HK(n, {\mathcal L})$ be the associated Hirzebruch–Kummer covering of exponent $n$. Rigidity conjecture. If ${\mathcal L}$…

Let $\mathcal F$ be a locally finite covering of the plane $\mathbb R^2$ by closed unit discs. Let $A,B$ be points in the doubly covered region, let $d$ be their distance, and let…

Let $\mathcal F$ be a locally finite covering of the plane $\mathbb R^2$ by closed unit discs. The doubly covered region consists of the points contained in at least two discs of…