24 problems

Alfaro's conjecture. If $A=B$, then a shortest $(A,B)$-network has no Steiner points and is a union of cycles.

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 $\mathcal{Q}_N$ 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…

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

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…

Superiority conjecture.

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 $d$ 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…

Let $k\geq 1$ and let $d$ satisfy … A good binary tree of height $k$ is defined recursively: a height-zero tree is a single node, and for positive height at most one child-subtree…

Let $d\geq 1$, and consider finite point configurations of $d+1$ points in Euclidean space. Their Steiner ratio is the ratio of the cost of the optimal Steiner tree to the cost of…

Zero-condition conjecture. The conditions from the Clean Condition are satisfied only when $F(i)=0$.

Let a finite set of discs be given. A packing is triangulated saturated if it is both triangulated and saturated, meaning that its contact structure is a triangulation and no furth…

Extremal great-circle distribution conjecture.

Monotonicity conjecture. For each fixed positive integer $p$, the sequence $R_p(n)$ is non-increasing in $n$.

Let $v$ be a root vertex with children $\{v_1, \dots, v_k\}$, where $k \geq 2$, the angles between children with respect to $v$ are at least $60^{\circ}$, and all children have rad…

Convexity conjecture. For any $n$, the least-area $n$-hedral tile is convex.

Let $\mathbb S^{n-1}$ be the unit sphere, and consider $N=n+1$ spherical caps of radius $R$ on it. The spherical-cap simplex conjecture. For every $R>0$, the maximal area of the un…

For $N=n+1$, let $v_1,\ldots,v_N\in\mathbb R^n$ satisfy $|v_1|=\dots=|v_N|=r$, and let $B_{v_i}(R)$ denote the ball of radius $R$ centered at $v_i$. The union-of-balls simplex conj…

Let an $\varepsilon$-approximate Steiner tree be a tree whose angles at Steiner points lie in $[2\pi/3-\varepsilon,2\pi/3+\varepsilon]$. For $d\geqslant 2$, let…

Triangular-row conjecture. For most $n$, there is an optimal configuration with $N$ rows. The $j$th row has $j$ points for $j\leq J$. If $n>N(N+1)/2$, the rows with $j>J$ each have…

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…