113 problems

A flexible polyhedron is a polyhedron whose faces remain rigid while its dihedral angles vary continuously. During such a flexion, its enclosed volume may in principle change. Bell…

For a unit vector $cvec{a}=(a^1,a^2,a^3)$ and $s oindenttext{in}cmathbb{R}$, let … be the totally umbilic planes in the upper-half-space model $(cmathbb{R}^3_+,b)$ of hyperbolic…

A Platonic solid is a regular convex polyhedron, and its dual Platonic solid is the Platonic solid obtained by interchanging faces and vertices. For each Platonic solid $P$, let…

Vinberg's polyhedral-inequality conjecture. There exist a family of subsets $M_{i}\subset\{1,\dots,N\}$ and a family of elements $l_{i}\in\mathfrak{t}_{\mathbb{Z}}$ such that the s…

Let $P\colon K\to\mathbb{R}^n$ be a flexible, not necessarily embedded polyhedron, and let $V_K(P)$ denote its generalized oriented volume, defined by integrating its characteristi…

Let $d\ge 2$ be a natural number, let $P$ be a $d$-dimensional polytope in $\mathbb{R}^d$, and let $p_0(x),\ldots,p_{d-1}(x)$ be polynomials. Bosse–Grötschel–Henk conjecture. There…

Let a polyhedron have edge lengths whose squares generate a ring, and let $V$ denote its volume. Connelly–Sabitov integrality conjecture. Both Connelly and Sabitov conjectured that…

A polytope is a polyhedral billiard table, and a knot type is an ambient-isotopy class of knots represented by a ball trajectory in that table. Infinite-knot-type conjecture. Any p…

A knot type is an ambient-isotopy class of knots in three-dimensional space, and a polytope is a three-dimensional polyhedral billiard table in which a ball follows a billiard traj…

Let $G$ be of type $A_n$, let $\underline{w}_0$ be any reduced decomposition of the longest Weyl-group element, and let $\lambda$ be a weight. Write $Q_{\underline{w}_0}(\lambda)$…

Kirillov's integrality conjecture. The polytopes $\mathrm{GT}_{\lambda,\beta}$ are integral polytopes.

Let $m,n$ be positive integers, let $\xi^m_n$ denote the generator indexed by $m$ inputs and $n$ outputs, and let $\partial$ be the perturbed differential in the minimal model of t…

Polyhedral semistable reduction conjecture. There exists a projective alteration $\Delta_B^1\to\Delta_B$, with induced alteration $\Delta_X^1\to\Delta_X$, and a projective subdivis…

Neoplatonic homotopy conjecture. Every $6$-net $\sigma$ has a family of realizations $\Delta(\sigma,l)$, each unique up to isometry, as undented hyperbolic polyhedra of side length…

Volume-max conjecture. For a prime $6$-net $\sigma$, its ideal neoplatonic realization maximizes volume among all ideal geodesic $2$-cycles with combinatorics $\sigma$.

Ideal-prime conjecture. If the $6$-net $\sigma$ is prime, its ideal neoplatonic realization $\Delta(\sigma,\infty)$ is convex.

Ideal neoplatonic conjecture. Every $6$-net $\sigma$ has a realization $\Delta(\sigma,\infty)$, unique up to isometry, as an ideal neoplatonic.

Neoplatonic conjecture. Any $6$-net $\sigma$ has a realization, unique up to isometry, as an undented Euclidean polyhedron $\Delta(\sigma)$ built from equilateral triangles of side…

Let $G$ be a finitely generated group satisfying the Atiyah conjecture, and let $C_*$ be a finitely generated free $L^2$-acyclic $\mathbb{Z}G$-chain complex. For a cohomology param…

Consider the problem of minimizing the first Dirichlet eigenvalue of the Laplacian among polyhedra, with the relevant geometric constraints as in the source. A simple polytope is a…

Minimal vertex-count conjecture. (i) For a CCP of a topological torus without self-intersection, the minimal number of vertices is $9$. (ii) For an orientable CCP of genus $g$, if…

Thin-ray conjecture. The function $f$ is unbounded below on $\mathcal P\cap\mathbb Z^n$ if and only if there exists a ray $\operatorname{ray}(y,d)$ of $\mathcal P$ such that, for e…

Let $psi\to z(psi)$ be the PLS-related mapping, and let $mathcal{C}z=lbrace psi otinmathbb{R}+^mbigm|V^T Zpsi=0rbrace$ be the associated polyhedral cone. Write its extremal rays as…

An embedded polyhedron is a polyhedral surface realized in Euclidean space without self-intersections. Rigidity conjecture for embedded polyhedra. Embedded polyhedra were conjectur…

Let $P$ be a triangulated polyhedron with distinguished vertices $A,B,A',B'$ satisfying … Form the twinned polyhedron $T$ of $P$ around the equator $ABA'B'$, as in the construction…