23 problems

Makeev's conjecture. There exists an $A\in SO(n)$ such that all $n(n+1)/2$ of these $(n-1)$-planes are concurrent. The conjecture is a reformulation of Makeev's universal-cover con…

Meylender–Thalmaier conjecture. The systolic ratio is strictly less than $1$ for every convex shape $K$, and it is maximized with respect to $K$ if and only if $K$ is a translate o…

Let $B$ be the unit disk in the plane, and for a convex shape $K\subset\mathbb{R}^2$ let $c(K\times B)$ be the length of the shortest closed generalized billiard orbit in $K$. Mall…

Let $\Gamma$ be a conical grid of order $n$. The covering number with multiplicity $k$ is the minimum number of lines required to cover every point of $\Gamma$ at least $k$ times.…

Levi-Hadwiger illumination conjecture. We have

Let $K_n^{(k)}$ be the complete $k$-uniform hypergraph, with its edges coloured using $r$ colours. A collection of cycles covers $(1-o(1))n$ vertices if the number of uncovered ver…

Soltan's homothetic covering conjecture. Assume that $K \in \mathcal{K}^d$ and that $\lambda_1 K, \ldots, \lambda_n K$ permit a translative covering of $K$ with…

Generalized affine plank conjecture. Assume that the convex sets $C_1, \ldots, C_n \subset \mathbb{R}^d$ permit a translative covering of the convex body $B \in \mathcal{K}^d$. The…

Let $Q_n$ be the $n\times n$ chessboard, and let $p_n$ denote the minimum number of lines whose union intersects every cell of $Q_n$. Piercing-number conjecture. For all $n \geq 3$…

Hadwiger's covering conjecture. For each $K\in\mathcal{K}^n$, $c(K)$ is bounded from above by $2^n$, and this upper bound is attained only by parallelotopes.

Bounded-color cover conjecture. Every such coloring has a monochromatic cover of order at most $r-1$ in which the subgraphs use at most $\lceil r/2\rceil$ distinct colors.

Generalized Hadwiger conjecture. For every $d$-dimensional convex body $B$ and every integer $\alpha\in[0,d]$,

Generalized Soltan conjecture. For all natural $d$ and all $\alpha$ such that $0\leq\alpha\leq d+1$,

Soltan's conjecture. For every natural number $d$,

Bezdek–Schneider's conjecture. The total inradius of the domains is at least $r$.

Spherical Goodman conjecture. The given caps can be covered by a cap of spherical radius

Spherical-segment covering conjecture. The total width of the spherical segments is at least $\pi$.

Let $P\subseteq\mathbb R^d$ be a polytope with $m$ facets. A covering by smaller-diameter sets is a cover of $P$ by sets whose diameters are strictly smaller than the diameter of…

Let $B^n$ be the Euclidean unit ball, and let $Z_1,\ldots,Z_N$ be $m$-dimensional Euclidean cylinders covering $B^n$. Write $\sigma_{n-m}(Z_i)$ for the $(n-m)$-dimensional volume o…

For a convex body $S\subset\mathbb C$, a symplectic cylinder is a set $S\times\mathbb C^{n-1}\subset\mathbb C^n$ or any of its images under symplectic transformations. Let…

Makeev's conjecture. Every convex body $K$ in $\mathbb{R}^d$ of diameter $\leq 1$ can be covered by a translate of $R_\Delta$ for some non-degenerate simplex $\Delta$ of diameter…

Bezdek–Bezdek relative-width conjecture. If ${\bf K}$ is covered by planks ${\bf P}_1,\dots,{\bf P}_n$ in ${\mathbb E}^d$, then

Let $d\geq 1$, and let $K$ be a convex body in $\mathbb R^d$, meaning a compact convex set with non-empty interior. A translate of $\lambda K$, with $0<\lambda<1$, is a smaller pos…