5 problems

Let $v_1,v_2,\ldots,v_n$ be unit vectors in a real Hilbert space $\mathcal{H}$. Fejes Tóth's real Hilbert-space zone conjecture. There exists a unit vector $u\in\mathcal{H}$ such t…

Let $C$ be a convex body, and let $P_1,\ldots,P_n$ be planks covering it. If $H_i$ is parallel to the boundary hyperplanes of $P_i$, define the relative width of $P_i$ by…

Bezdek's conjecture. The total width of the planks is at least $1$.

Let $\mathbf{K}$ and $\mathbf{C}$ be convex bodies in $\mathbb{E}^d$, with $d\geq 2$, and let $m$ be a positive integer. For each body, let $r_{\mathbf{C}}(\cdot,m)$ denote its $m$…

A zone is the spherical neighbourhood of a great circle, with width equal to twice its maximal spherical distance from that great circle. Let zones of widths $w_1,\ldots,w_n$ cover…