210 problems

Let $D\subsetneq\mathbb{R}^n$ be a convex domain, let $x\in D$, and let $r\leq \arctan(\sqrt{2})/\sqrt{2}$. Define the quasihyperbolic metric by…

For every metric-measure space $(X,d,\mathfrak m)$, every $K\in\mathbb R$, and every $1<N<\infty$, is $\mathrm{BCD}(K,N)$ equivalent to $\mathrm{RCD}(K,N)$? Equivalently, does the…

Let $X$ and $Y$ be metric surfaces, and let $f:X\to Y$ be an area-preserving Lipschitz map. Does there exist a constant $K\ge 1$ such that, for $2$-modulus-almost every rectifiable…

For each $\alpha\ge 1$, let the $\alpha$-Grushin plane be $\mathbb{R}^{2}=\mathbb{R}_{x}\times\mathbb{R}_{y}$ equipped with Lebesgue measure and the sub-Riemannian structure genera…

For fixed $n\ge 3$ and $c\in(0,1)$, let $\mathcal{GT}(n,c)$ be the class of geodesic metric trees of valence at most $n$ whose branch points are uniformly relatively separated with…

For $1\le p<\infty$, even $m\ge 4$, and $n\in\mathbb N$, let $A_p(n,m)$ be the least constant such that every function $f:\mathbb Z_m^n\to L_1$ satisfies … Determine the optimal or…

Let $\mathcal{A}$ be the class of complete Alexandrov spaces of nonnegative curvature. For integers $n\ge 1$ and dyadic $m\ge 2$, define $\Gamma_{\mathcal{A}}(n,m)$ to be the least…

Let $X$ be a Hadamard space and let $r=\operatorname{asrk}(X)$ be its asymptotic rank. For every integer $k\ge r$, there exists a constant $C=C(X,k)>0$ such that every integral $k$…

Let $X$ be a normed linear space, let $C\subseteq X$ be nonempty, and let $T:C\to C$. Is it true that if there exists $b\geq 0$ such that…

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…

Triangle inequality conjecture. For any pure states $\textup{\textbf{x}},\textup{\textbf{y}},\textup{\textbf{z}}\in\mathbb{P}(\mathbb{C}^n)$,

Let $D54A^m$ denote the unit $m$-sphere with its intrinsic spherical metric, let $d_{\operatorname{GH}}$ be the Gromov–Hausdorff distance, and let $D701_m$ be the quantity defined…

A game space is a compact geodesic metric space equipped with the Cops and Robber game. Its doubling constant is the least $k\in\mathbb{N}\cup\{\infty\}$ such that every ball…

Let $(V,d)$ be a finite metric space with $n$ points. For distinct points $x,y,z\in V$, say that $z$ lies between $x$ and $y$ when … For distinct $x,y\in V$, define the line genera…

Let $\gamma_{2k+1}:\mathbb{S}^1\to\mathbb{S}^{2k+1}$ be the odd trigonometric moment curve embedding, let $\Pi_{2k+1}:\mathbb{S}^{2k+1}\to\gamma_{2k+1}(\mathbb{S}^1)$ be the neares…

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…

Nabutovsky's conjecture. There exist geodesic nets in the Euclidean plane with $4$ unbalanced vertices and an arbitrarily large number of balanced vertices. Moreover, this may alre…

Edge-contraction and subdivision conjecture. For all $L,C\in\mathbb{N}$, there exists $C'\in\mathbb{N}$ such that if a graph $G$ is $(L,C)$-quasi-isometric to a graph in…

Let … be a non-collapsed Gromov–Hausdorff-convergent sequence of compact Alexandrov spaces of dimension $n$ with curvature bounded below by $-1$. Perelman's bi-Lipschitz stability…

Let $n$ be the ambient dimension, let $u,v\in\mathbb{R}^n$ be distinct points, and let $[u,v]$ denote the line segment joining them. For a parameter $c$, write…

The conjectured Lipschitz bound. For all $x,y,a\in\mathbb{B}^n$,

Strong hull property theorem. The graph $\operatorname{Cay}(W)$ has the strong hull property.

Alexandrov gluing conjecture. The gluing produces an Alexandrov space $Y$ if and only if the gluing is by isometry and, for every $p\in Y$, $T_p(Y)$ is a metric cone over…

Scaling-limit conjecture. There exists a random compact metric space $\mathcal{L}^{(S)}_{\delta}$ such that