19 problems

Let $K$ be a convex body in $\mathbb{E}^3$. The density of a packing is the proportion of space occupied by the translates of $K$, and the densest packing density of $K$ is the sup…

Let $\mathcal{K}$ be a finite family of pairwise intersecting convex sets in $\mathbb{R}^3$. Constant-line transversal conjecture. The family $\mathcal{K}$ admits a transversal by…

Let $\mathcal{K}$ be a finite family of $n$ convex sets in $\mathbb{R}^3$, and suppose that every two members $K_1,K_2\in\mathcal{K}$ satisfy $K_1\cap K_2\neq\emptyset$. Fractional…

Let $\mathsf{ConjInv}$ denote the conjugation invariants in dimension $d=3$, and let a shuffle with letters mean a shuffle product involving letters. The signed volume is the degre…

Fractional line-piercing conjecture. There exists a constant $c$ such that some line intersects at least $c|\mathcal{F}|$ members of $\mathcal{F}$.

FCC truncated-density conjecture. For all $>0$, among packings of unit balls in $^3$, the $$ -truncated density has a maximum at the corresponding FCC lattice.

For a finite point set in Euclidean three-space, define distinct triangles by congruence classes of triples of distinct points. Three-dimensional three-triangle conjecture. The max…

For $d\in\{2,3\}$, let $A_{\mathrm{gen}}^{(d)}(n)$ denote the minimum number of distinct angles formed by a set of $n$ points in general position in $\mathbb R^d$, where general po…

Let $E\subset\mathbb R^3$ be a large finite point set of $n$ points. An angle 2-chain consists of three consecutive angles determined by four points, and $\Lambda_2(E)$ denotes the…

Minimally rigid graphs in ${\mathbb R}^3$ are constructed from the complete graph $K_3$ by Henneberg moves H1, H2, and H3. Henneberg-step conjecture. H1, H2, and H3 completely char…

Let $v_{ab}\in S^2$ be the unit vectors associated with a configuration of four distinct points in $\mathbb{R}^3$, let $D$ be the normalized Atiyah-Sutcliffe determinant, and let…

Let $C_n(\mathbb{R}^3)$ be the configuration space of $n$ distinct points, and for $a\ne b$ let $v_{ab}=(\mathbf{x}_b-\mathbf{x}_a)/\|\mathbf{x}_b-\mathbf{x}_a\|\in S^2$. Write…

Let the face-centered cubic and body-centered cubic lattices in ${\mathbb R}^3$ be scaled to the specified density $\rho$, and let the potential be $r \mapsto e^{-\pi r^2}$. Gaussi…

No further 8-rep-tile tetrahedra conjecture. There are no further $8$-rep-tile tetrahedra.

Let $d$ satisfy $d\in(\sqrt{2},\sqrt{2}+\varepsilon]$, where $\varepsilon>0$ is sufficiently small, and consider translates in ${\Bbb R}^3$ of a string of unit balls whose centres…

For a finite set $P$ of $n$ points in three-dimensional space, let $D(P)$ denote the number of distinct distances determined by pairs of points of $P$. Distinct-distances conjectur…

Invar-list continuation conjecture. The process of selecting scalar curvature invariants that cannot be written as polynomials of the preceding invariants can be continued for the…

Let $G=(V,E)$ be a generically 3-isostatic graph, and let $X\subseteq V$ satisfy … Let $e$ and $f$ be distinct edges whose vertices lie in $X$. An implied $K_4$ is a complete graph…

Many-hole domain conjecture. If $k$ is sufficiently large and