155 problems

Let $S^{d-1}\subset\mathbb R^d$ be the unit sphere, and let zones be centrally symmetric parts of the sphere covered by planks. Fejes Tóth's conjecture. If finitely many zones cove…

Let $(M,g)$ be a closed orientable Riemannian three-manifold, with $g$ a metric of constant sectional curvature $\pm1$. Let $\operatorname{Isom}(M)$ be the isometry group of $M$ an…

Let $M$ be a compact oriented $3$-manifold with finite fundamental group. A spherical $3$-manifold is a quotient $\mathbb{S}^3/G$, where $G$ is a finite subgroup of…

Fejes Tóth's conjecture. This energy is maximal when $x_1,\dots,x_{d+1}$ are mutually orthogonal and $x_k=x_{k-(d+1)}$ for every $k$ with $d+1<k\le N$. Equivalently, periodically r…

Consider partitions of the sphere into three cells, with the objective given by the sum of their first Laplace--Beltrami eigenvalues. Let the $Y$ partition be the partition consist…

Let $\mathbb{B}^3$ be the Euclidean unit ball, and let an embedded free boundary minimal annulus be an embedded minimal annulus whose boundary meets $\partial\mathbb{B}^3$ orthogon…

Let $P(u)$ be a flexible polyhedron in the sphere $\mathbb{S}^n$, where $n\ge 3$. Replacing a vertex by its antipode means replacing that vertex of $P(u)$ with its antipodal point…

Let $q$ satisfy $2\leq q\leq n+2$, and let $\Omega$ be a $q$-cluster on $\mathbb{S}^n$ with prescribed volume $V(\Omega)=v\in\operatorname{int}\Delta^{(q-1)}_{1}$. A standard bubbl…

Symmetry-count sufficiency conjecture. For all such groups $\Gamma$, the counts in Corollary are sufficient for $(G,\chi,p)$ to be $\Gamma$-symmetric isostatic.

Approximation conjecture. Any spherical convex body of constant width $\pi/2$ can be approximated by a sequence of spherical convex polytopes of constant width $\pi/2$.

Let $S^{n-1}$ be the unit sphere in Euclidean $n$-space, and let $A\subset S^{n-1}$ be a measurable set. Call $A$ orthogonality-free if it does not contain two orthogonal vectors.…

Let $e_3$ be the third standard basis vector, let $\mathbb{S}^2$ be the unit sphere, and consider the class $\mathcal{P}_{3,K}\cap\mathcal{I}_3$ of polytopes with $K$ vertices. For…

Consider the optimization problem for five points on a sphere under the potential $\operatorname{sign}(s)/x^s$, where $s$ is the parameter. Let the triangular bipyramid be the conf…

Let $T$ be a spherical tetrahedron whose edge lengths are $l_{ij}=\pi\alpha_{ij}$ for $1\leq i<j\leq4$, with associated dihedral angles $\theta_{ij}$, volume $V$, and spherical Gra…

Let $P$ be an embedded closed polygon in $S^2$ that bisects the area, and call a vertex configuration a flattening when it has the polygonal analogue of an inflection point. Discre…

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…

Let $S$ be a hyperbolic or spherical $n$-simplex, and let its volume, viewed as a function of the dihedral angles, be continuously extended to degenerated simplexes. Let the closur…

Let $S$ be a hyperbolic or spherical $n$-simplex, and regard its volume as a function of its dihedral angles. The space of such simplexes may degenerate, yielding a closure of the…

Möller's bound conjecture. A version of the bound in Corollary should hold for every centrally symmetric subvariety $X\subset S^n$, even when the measure $\mu$ is not centrally sym…

Let $csigma_{n-1}$ be a Haar measure on the unit sphere $S^{n-1}$. Let $A$ be a symmetric subset of $S^{n-1}$, and let $P=\{x\in S^{n-1}:|x_{1}|\leq t\}$ be a symmetric spherical s…

Let spherical polyhedra in $cmathbb{S}^3$ be considered up to scissors congruence, and let their volume and Dehn invariant be the corresponding invariants. Spherical scissors-congr…

Let $\mathbb{S}^n$ be the unit sphere in $\mathbb{R}^{n+1}$, with spherical $n$-measure $\mu_n$. Let $\mathcal C_n(w)$ denote the class of spherical convex bodies of constant width…

Let $T_{d,q}(\lambda,L;C)$ denote the maximal cardinality of an admissible set of lattice points on $\lambda S^{d-1}$ with the parameters $d,q,\lambda,L,C$. Transverse-configuratio…

Let $A_{k,d,\lambda}$ denote the maximal number of lattice points in the intersection of $k$ unit-width, $\nu$-transverse bands on the sphere $\lambda S^{d-1}$, with…

Let $d\geq 2$, let $u_1,\dots,u_n\in\mathbb{S}^{d-1}$, and let $\alpha\in[0,\pi/2]$. For a unit vector $u$, define the spherical zone … For $j=1,\dots,n$, set … Let $\sigma_{d-1}$…