65 problems

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…

Let $\operatorname{\mathcal P}(\mathbb X)$ be the scissors congruence group of polytopes in a 3-dimensional geometry $\mathbb X$, and let … be the kernel of the Dehn invariant. Deh…

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 $A_{k,d,\lambda}$ 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 $\sigma_{d-1}$…

Admissibility criterion. A regular spherical polygon is admissible if and only if the inner angle of its polar is at least

Let $\mathbb{S}^2$ be the unit $2$-sphere. Spherical coloring conjecture. There exists a map $q:\mathbb{S}^2 \rightarrow \{\pm 1, \pm 2, \pm 3, \pm 4\}$ such that antipodal points…

Let $R\in[0,\frac{\pi}{2}]$ and let $\mathbb{M}^3_{\kappa_R}$ be the space form of curvature $\kappa_R=\sin^2 R$. An embedded free boundary minimal annulus in…

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…

For each $3$-connected embedded graph $(G,\mathbf p)$ on the unit sphere with topological degree one and with every facial perimeter of length less than $2\pi$, define a partial or…

Let $G$ be a vertex $3$-connected planar graph embedded in the unit-radius sphere. Assume that each edge is a geodesic of length less than $pi$ and each face has total perimeter le…

Let $\mathbb{S}^{(d-1)}$ be the sphere and let $\Delta:\mathbb{S}^{(d-1)}\to [0,1]$ denote the spherical cap discrepancy. Global Lipschitz continuity conjecture. The spherical cap…

Let $p>0$, and for a probability measure $\mu$ on $\mathbb{S}^{d-1}$ define the $p$-frame energy by … A minimizer is a probability measure attaining the minimum of this energy. Bil…

Let $\mathbb{S}^2$ be the unit sphere and let $\mathcal{E}_2(\mathbb{S}^2;N)$ denote the minimal Riesz $2$-energy of $N$ points on $\mathbb{S}^2$. Let $\gamma_n(\alpha)$ be the gen…

Let $n\ge 4$, and consider exotic flexible cross-polytopes in the sphere $\mathbb{S}^n$. These are flexible cross-polytopes whose dihedral-angle tangents have the exotic parametriz…

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…

Integral Fejes Tóth conjecture. The maximum of $I_1(\mu)$ over all Borel probability measures on $\mathbb S^d$ is achieved by $\mu_{ONB}$. The discrete conjecture remains open for…

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…

Let $F4D0\mathbf{Z}_1,\dots,\mathbf{Z}_n\subset \mathbb{S}^{d-1}$, with $d>2$, be zones of widths $2\beta_1,\dots,2\beta_n$, respectively. Suppose that … consists of $2m$ spherical…

In spherical coordinates on $\mathbb{S}^2$, define … and … where $(x(\phi,\theta),y(\phi,\theta),z(\phi,\theta))=(\cos(\phi)\sin(\theta),\sin(\phi)\sin(\theta),\cos(\theta))$. Let…

Let $\sigma^8$ be the space of homotopy Hopf manifolds, let $\widetilde{\sigma}$ be the map appearing in the source, and let $\theta^7$ denote the group of homotopy seven-spheres.…

Numerical-map conjecture. There is a numerical map $N$ which is constant with value $\frac{1}{12}$ on $\sigma^8_{\textrm{Milnor}}$.

Let $\Omega$ be a strictly convex domain with smooth boundary $M$ in $\mathbb{S}^{n+1}$. For integers $k$ and $l$ satisfying $0\leqslant k\leqslant n$ and $0\leqslant l\leqslant k$…

Let a non-Kochen–Specker set in dimension $d$ be a measurable subset $A\subseteq S^{d-1}$ admitting a valuation map $v:A\to\{0,1\}$ such that $v(-n)=v(n)$, every mutually orthogona…

Let $A\subseteq S^2$ be a measurable subset containing no two orthogonal directions. Its measure is normalized by $|S^2|=1$. Kalai–Wilson's conjecture. The maximal measure of such…