59 problems

A minimum spanning tree is a shortest connection of a finite set $A$ of points in the plane by segments with endpoints in $A$. The Steiner ratio is the infimum, over finite point s…

Let $M$ be a circle of radius $R$ and let $r$ satisfy $R>r$. A maximal distance minimizer is a connected compact set of minimal length whose closed $r$-neighborhood contains $M$. M…

Let $\rho_2$ denote the infimum, over finite point sets in the Euclidean plane, of the length of a Steiner tree divided by the length of a Euclidean minimum spanning tree. Gilbert–…

Let $\mathcal{M}$ be a normal, convex mosaic in $\mathbb R^3$ whose cells have unit volume, and let $\underline{\rho}_1(\mathcal{M})$ denote its lower edge density. Lower edge-dens…

Let a moving sofa be a connected planar shape that can be moved around a right-angled corner in a hallway of unit width. Let $\mu_\textrm{G}=2.21953166\ldots$ denote Gerver's const…

Let $\operatorname{ssr}(\mathbb R^2)$ denote the Steiner subratio of the Euclidean plane, defined as the infimum of the ratios of minimal-filling weight to Steiner minimal-tree len…

Let $\mathbb{M}^n$ be a piece-wise differentiable, elliptic Minkowski space, and let $s(\mathbb{M}^n)$ denote the maximum degree of a Steiner point in a Steiner minimal tree in…

Let $\mathbb{M}^n$ be an $n$-dimensional Minkowski space, and let $s(\mathbb{M}^n)$ denote the maximum degree of a Steiner point in a Steiner minimal tree in $\mathbb{M}^n$. Morgan…

Cieslik's conjecture. The maximum degree satisfies

Let $\mathcal{Q}_N$ denote the configuration space of $N$ points in three-dimensional space, and let $F_N$ be the angular sum defined in the paper. The asymptotic spherical-distrib…

Let $\mathcal{Q}_N$ denote the configuration space of $N$ points in $d$-dimensional space, and let $F_N$ be the angular sum defined in the paper. The eight-point maximum conjecture…

The five-point maximum conjecture.

Let $n$ be the number of congruent non-overlapping circles packed in a rectangle, and let $L/S$ denote the rectangle's aspect ratio, where $L$ and $S$ are its longer and shorter si…

Morgan's conjecture. For all $d\geq 2$,

Let $\Delta(k)=\frac{k(k+1)}{2}$, and consider packings of equal nonoverlapping disks in an equilateral triangle. Let $d(n)$ denote the largest possible minimum distance between th…

Let $Q$ be the region and let $P=(p_1,\ldots,p_n)$ be the generator locations. The multi-incenter objective is … where $V_i$ are the Voronoi cells of the generators. Sphere-packing…

Hexagonal packing conjecture. For $n<54$, the minimum perimeter is achieved, perhaps nonuniquely, by a subset of the hexagonal circle packing.

Let a Steiner network have terminal nodes, Steiner nodes, and entry/exit points on the boundaries of soft obstacles. After perturbing the terminal nodes, consider the positions of…

A compact convex set $S\subseteq\mathbb{R}^3$ is Rupert if a second identical copy can pass straight through a hole in the interior of the first with rescaling factor $\mu_S>1$. St…

For a compact convex set $S\subseteq\mathbb{R}^3$, its Nieuwland constant $\mu_S$ is the largest rescaling factor for which a second copy of $S$ can pass straight through a hole in…

Let $\Omega$ be a triangle with vertices $A$, $B$, and $C$. Assume that its diameter is the side $[AB]$ and that $\widehat{CAB}\geq\widehat{CBA}$. For $c\in[0,|\Omega|]$, consider…

Consider a searcher in the entire plane looking for a target whose distance and direction are unknown. A search strategy or path is evaluated by the distance or search cost require…

Let a convex regular polygon be given as the forest, and suppose the hiker does not know the starting point or facing orientation. An escape path is evaluated by the worst-case dis…

Superiority conjecture.

Let $d$ be the number of terminals in a regular simplex, and let $k\geq 1$. The construction described immediately before the claim recursively splits Steiner points and coordinate…