201 problems

Let $T$ be a finite metric tree of total length $L$, and let $r>0$ satisfy $L\ge 2r$. Then there exists a probability distribution on $0$-good ball covers of $T$, with all ball rad…

For every integer $n\geq 1$ and every real $p$ satisfying $2<p<\infty$, the maximum cardinality of an equilateral set in $\ell_p^n$ is $e(\ell_p^n)=n+1$. Equivalently, every set…

Homeomorphism extension. The homeomorphism version of Theorem would also hold for any thin point $x$ with $f_*={\rm id}$ and for any $x\in X_0^2$ if the ball…

Let $a,b,c,d$ be four complex points on the unit circle $S^1$ in this order, with the Euclidean lines $L(a,b)$ and $L(c,d)$ nonparallel. Let $h$ be an arbitrary point on the Euclid…

Lytchak's Regular Point Conjecture. The Tits boundary $\partial_T X$ contains a regular point.

Metric thickening finite-support homotopy equivalence conjecture. The inclusion induces a homotopy equivalence

Let $(X,\mathsf d,\mathcal{H}^{N})$ be a non-collapsed $\mathsf{RCD}(K,N)$ space, and let $({\mathcal R}_{N})_{\varepsilon}$ denote its $\varepsilon$-regular set. Then the interior…

Let $(X,d)$ be a Busemann $G$-space, meaning that it is a metric space satisfying Menger convexity, finite compactness, local extendibility, and uniqueness of extension. Suppose th…

Large-cohomogeneity diameter conjecture. For every $\epsilon>0$ and all sufficiently large $k$ with $n>k$,

Let $(\mathbb H,d^{\mathbb H})$ be the Heisenberg group equipped with its Carnot–Carathéodory metric. Lee–Naor conjecture. The metric space $(\mathbb H,d^{\mathbb H})$ does not adm…

Berestovskis conjecture. The space $X$ is homeomorphic to the topological product

Distance-realization property. Then the distance $\operatorname{dist}(P,S_R)$ is realized by the point

Let $\mathcal{M}$ be the family of source spaces and $\mathcal{Y}$ the family of pointed target spaces. Suppose that a net $M_i\in\mathcal{M}$ converges in the measured Gromov–Haus…

Uniform lattice conjecture. Every lattice $\Gamma\subset M$ with the $\mathcal{LE}$ is uniform. The source places this definition immediately before the conjectural discussion, but…

Orbit lattice conjecture for platycosms. All 10 platycosms have the orbit lattice property.

Lipeomorphic invariance conjecture. Metric-coordinate dimension is invariant under lipeomorphisms; that is, if $M$ and $N$ are lipeomorphic, then they have the same metric-coordina…

Let $X$ be a non-degenerate compact metric space, and suppose that every midset of $X$ is homeomorphic to the $(n-1)$-sphere $S^{n-1}$. Higher-dimensional double midset conjecture.…

A continuum is a non-degenerate connected compact metric space. For a metric space, the double midset property (DMP) means that every pair of distinct points has a midset consistin…

Let $V$ be a Kähler manifold, let $\mathcal{H}$ be the space of Kähler potentials, and let $\mathcal{H}_0$ denote its normalized subspace. For a normalized $\phi\in\mathcal{H}_0$,…

Metric cone conjecture. If the Kähler–Ricci shrinker has maximal volume growth, then $(Z_0,d^Z_0)$ is a metric cone and coincides with the metric completion of…

Let $(\mathsf{X},\mathsf{d}_{\mathsf{X}})$ be a compact length metric space with finite Hausdorff dimension. A bounded-length-distortion map, or BLD map, is a continuous map into a…

Optimality conjecture. The constant $m=\frac{2}{n+1}\left(\frac{n}{n+1}\right)^n$ is optimal for all Kähler manifolds $(X,\omega)$.

Projected-stationarity conjecture. For every $H\geq 2$, the complete Das–Dennis grid $G_H$ is projected-stationary for anchored-box magnitude on $\Delta_2$. More precisely:

Let $d$ be a metric of strict negative type on a finite space in general position, let $\mathcal{I}$ be a subset of $\operatorname{peel}(d)$, and let $d|_{\mathcal{I}}$ denote the…

Let $d$ be the metric induced on a finite subset of $\mathbb{R}^m$, let $\mathcal{I}$ be a subset of $\operatorname{peel}(d)$, and let $d|_{\mathcal{I}}$ denote the restriction of…