20 problems

Let $G_n$ be a family of bipartite graphs with $|G_n|\to\infty$, each having maximal degree $d$, where $d$ is independent of $n$. For vertices $v_n\in G_n$, let $f_n$ be uniformly…

For a subset $S$ of a metric space $X$, let $\lambda(S,X)$ denote the norm of a bounded linear extension operator from Lipschitz functions on $S$ to Lipschitz functions on $X$, and…

Let $U$ be a bounded open subset of $\mathbb C$, let $x\in U$, let $t=0$, and let $\phi$ be an admissible function. Let (a) denote the full-area-density condition and (c) the conve…

Let $U$ be a bounded open subset of $\mathbb C$ with $x\in U$, let $Y$ be the closure of $U$, let $t$ be a non-negative integer, and let $\phi$ be an admissible function. Denote by…

Convergence conjecture.

Path-maximization conjecture. Among all $n$-vertex graphs, the maximum of

Let $M$ be a metric space, let $\operatorname{dens} M$ denote its density, and let $\operatorname{Lip}_0(M)$ be the Banach space of Lipschitz functions on $M$ vanishing at the dist…

Let $A\subseteq\mathbb{R}$ be an $F_{\sigma\delta}$ set. A continuous function $f\colon\mathbb{R}\to\mathbb{R}$ has lower pointwise Lipschitz constant … at precisely the points of…

Let the Sierpiński gasket carry Brownian motion, and let the domain of its Dirichlet form consist of the functions associated with that diffusion. A function is Lipschitz when it i…

Let $X$ be an $L_1$-predual Banach space, and suppose that $X$ has the finite-CEP, meaning that every Lipschitz map from a subset of a metric space into $X$ with finite image admit…

Lipschitz-function conjecture. Every non-constant Lipschitz function is not of bounded variation with respect to definition $(A)$.

Let $M$ be a metric space, let $Z$ be a Banach space, and let $\textnormal{Lip}_0(A,Z)$ denote the Banach space of Lipschitz maps vanishing at the base point. Define … where … and…

Tangent-field characterization conjecture. Sets of non-differentiability of real-valued Lipschitz functions may be described as precisely those sets for which there is an $(n-1)$-d…

Makai–Pach's local-density conjecture. If every unit ball centered at $x\in\mathbb{R}^m$ contains at most

The Lipschitz-surface characterization conjecture. The sequence $(x_i)_{i\in I}$ is $d$-controlling if and only if there exist a Lipschitz map $g:\mathbb{R}^d\to\mathbb{R}^m$ and a…

The density characterization conjecture. The sequence $(x_i)_{i\in I}$ is $d$-controlling if and only if

Let $f:{\mathbb{R}}^d\longrightarrow {\mathbb{R}}^m$ be continuous and piecewise Lipschitz with exceptional set $\Theta$. Suppose that $f$ is intrinsically Lipschitz with respect t…

Let $\mathbb{G}$ be a finitely generated group and let $\mu$ be a courteous probability measure on $\mathbb{G}$. Let $\mathsf{LHF}(\mathbb{G},\mu)$ denote the space of Lipschitz…

Extension-constant conjecture. If

Let $M$-Lipschitz functions on the two-dimensional lattice $\mathbb{Z}^2$ be sampled according to the random model under discussion. Flatness conjecture. Flatness no longer holds f…