89 problems

A hyperbolic strictly-convex polyhedron is a strictly-convex polyhedron in hyperbolic space, with dihedral angles measured along its edges. Stoker's conjecture. Every hyperbolic st…

For a polyhedron $P$, define its isoperimetric quotient by … where $V$ is its volume and $F$ its surface area. Two polyhedra are isomorphic when they have the same combinatorial ty…

Lattice-packing conjecture. The densest packings of the centrally symmetric Platonic and Archimedean solids are given by their corresponding optimal lattice packings.

Let $\pp$ be a compact, oriented, irreducible, atoroidal $3$-manifold with boundary equipped with an angled block structure, and let $\tilde{\pp}$ denote its universal cover. The a…

Milnor's continuity and vanishing conjecture. The volume function $V$ admits a continuous extension to $\overline{\Omega}$. Furthermore, the points on $\partial\Omega$ where $V$ va…

Stability-index conjecture. Every $n$-dimensional polytope can be represented by $n$ polynomial inequalities.

DDF's conjecture. For every even $n\geq 20$ with $n\ne 22$, there exists a tight $5_n$.

$4_n$ existence and zigzag conjecture. (i) A z-knotted $4_n$ exists if and only if $n\geq 30$ and $n\equiv 2\pmod 4$. (ii) A tight $4_n$ exists if and only if $n\geq 8$ and…

Symmetry-group conjecture. (i) $3_n(D_{2h})$ exists if and only if $\frac{n}{4}$ is even and $n\geq 16$; there are no tight $3_n(D_{2h})$. (ii) $3_n(D_{2d})$ exists if and only if…

Curvature-graph conjecture. The graph of curvatures of any tight graph of type $3_n$ is the graph in the first of those three cases.

DDF's conjecture. Every z-knotted $3$-valent plane graph has an odd number of edges of type I.

Let $P$ be a convex polyhedron in $\mathbb{R}^n$ and let $g_0$ be the Euclidean metric on $P$. For each codimension-one face $F_i$, write $H_g(F_i)$ for its mean curvature, and for…

Let $\mathcal{P}_a$ be the stellated tetrahedron with vertices … … … … where edges join all $p_i,p_j$ and $p_i,q_j$ for $i\ne j$, and $a\in(0.5,0.57)$. A polyhedron is Rupert if it…

Let $\mathcal{P}$ be a polyhedron. It is Rupert if there exist $R_1,R_2\in\operatorname{SO}(3)$ and $t\in\mathbb{R}^2$ such that … where $P:\mathbb{R}^3\to\mathbb{R}^2$ drops the…

Let $\mathcal{P}$ be a convex polyhedron, meaning the convex hull of a finite set of points in $\mathbb{R}^3$ in convex position. A polyhedron is Rupert if there exist…

Domokos–Horváth–Goriely–Regős conjecture. Every polyhedral tiling can be completely softened.

Duality conjecture. A convex hyperbolic c-polyhedron has a convex hyperbolic dual if and only if it has an orientation of its vertex-disks so that all inversive distances are in…

Global shallowness conjecture. A hyperbolic, locally shallow, convex c-polyhedron is globally shallow.

A Kochen–Specker (KS) set is a set of rays, or equivalently projectors, together with their orthogonality relations. Such a set is rigid if any set of projectors, not necessarily o…

A polyhedron is a solid bounded by finitely many polygonal faces, and an isometric deformation is a deformation preserving the intrinsic metric of the polyhedron. Bellows conjectur…

The rhombicosidodecahedron is the convex Archimedean solid formed by squares and triangles, and a polyhedron is called Rupert if a congruent copy of it can pass through a straight…

A convex three-dimensional polyhedron is called Rupert if a congruent copy of it can pass through a straight hole inside the original polyhedron. Jeřábek–Weiss–Yusun conjecture. Ev…

Finite-subcomplex relaxation conjecture. Condition (3) can be relaxed so that it is required to hold only outside a finite sub-complex of $\mathcal{D}$.

A 3-deformation is a sequence of elementary expansions and collapses involving cells of dimension at most $3$. A $2$-complex is contractible when it is homotopy equivalent to a poi…

Várkonyi's conjecture. The complexity satisfies