19 problems

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 $P$ be a metric space, and let $\big((M_x),\,(\mu_{xy})\big)$ be an approximate action of the pair groupoid $P\times P$, with each $M_x$ a complete metric space and $\mu_{xy}$…

Uniqueness conjecture. Every metric space $X$ with a closed and discrete set $\operatorname{dist} X$ has the property of uniqueness in the class $\operatorname{\mathcal{G\!H}}$.

Consider $d53d^n$ endowed with the metric $d(x,y)=\lVert x-y\rVert$, where the norm is any norm on $d53d^n$. Let $S_2$ be a subdivision scheme on $d53d^n$ satisfying, for…

Lipschitz magnitude conjecture. Let $T\subseteq L_1$ be tractable. Then

Completeness. With such a topology, $\mathcal{TA}\mathcal{F}(M)$ forms a complete metric space, or is complete in a relevant sense.

Recall that $\aleph_{1}$ denotes the first uncountable cardinal, and let $D_{\aleph_{1}}$ be the discrete space of cardinality $\aleph_{1}$. The space…

Let $\mathbf{M}$ denote the class of metric spaces, let $\mathbf{F}$ be the monoid of functions under composition, and let $\mathbf{P}_{\mathbf{X}}$ be the monoid associated with a…

Let $\mathbf{M}$ denote the class of metric spaces, let $\mathbf{P}_{\mathbf{X}}$ be the monoid associated with a metric space $\mathbf{X}$, and let $\mathbf{Am}$ be the monoid of…

Pointwise-convergence failure conjecture. The conclusion that $p$ is a Birkhoff-James extension fails if the assumption that $p(x_n,\,\rule{3mm}{0.15mm})$ and…

Let $X$ and $Y$ be Banach-space-valued random variables, and let $f$ be a similarity transformation of the Banach space. The Banach-space perfect-correlation conjecture. The axiom…

The optimal coupling conjecture. Under “general conditions”, the infimum can be attained or computed among couplings satisfying $X=X'$ or $Y=Y'$. The text presents this as a conjec…

Let $G=(V,E)$ be a connected graph with at least two vertices. Let $\ell(G)$ be the number of distinct graph-metric lines, and let a universal pair be a pair of vertices generating…

Let $G$ be a connected graph, let $\ell(G)$ be the number of distinct graph-metric lines, and let $\textsc{br}(G)$ denote the number of bridges of $G$. A pendant edge is an edge in…

The contractive-grid conjecture. Every such grid contains a 1-way Cauchy sequence. The conjecture is introduced as a formulation capturing the essence of Austin's common fixed poin…

Let $r\geq 3$ be an odd integer, and let $M_r(n)$ denote the class of finite metric spaces on $[n]$ with distances in the relevant $r$-element distance set. Let $\mathcal{L}_r$ be…

Let $(V,\rho)$ be an $n$-point metric space with $n\geq 2$, and suppose that the number of distinct distances occurring in the metric is bounded by an absolute constant. Conjecture…

A complete admissible metric $d$ for a $\sigma$-compact, locally compact space $X$ is a metric inducing the topology of $X$ and complete as a metric space. A Heine–Borel metric is…

Let $(X,d)$ be a metric space, and let $q_X$ denote its metric cotype exponent. Metric cotype gap conjecture. There is no metric space $(X,d)$ for which … The preceding results est…