4 problems

Uniform perfectness conjecture. Uniform perfectness of $\hat{\mathsf C}\setminus\Omega$ should be necessary for $(\Omega,k)$ and $(\Omega,h)$ to be quasi-isometrically equivalent.

Let $a\in\mathbb{R}^n$, let $D=\mathbb{R}^n\setminus\{a\}$, let $x\in D$, and let $B_k(x,s)$ and $B_c(x,s)$ denote quasihyperbolic and Cassinian metric balls, respectively. Sharp C…

Let $D\subsetneq\mathbb{R}^n$ be a proper subdomain, let $x\in D$, and let $\delta_D(x)$ be the Euclidean distance from $x$ to $\partial D$. For a metric $m$, write…

For $0<s<t$, let $G_s$ and $G_t$ be the planar domains … For points $z_1,z_2\in G_s$, write $k_{G_s}(z_1,z_2)$ and $k_{G_t}(z_1,z_2)$ for their quasihyperbolic distances in the res…