5 problems

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 $\mathbb{B}^n=\{x\in\mathbb{R}^n:|x|<1\}$, let $x\in\mathbb{B}^n$, and let $B_\rho(x,s)$ and $B_c(x,s)$ denote hyperbolic and Cassinian metric balls, respectively. Sharp Cassin…

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…

Let $G\subset\mathbb{R}^n$ be a domain that is starlike with respect to $x\in G$, and let $B_{c_G}(x,r)$ denote the Cassinian ball of radius $r$. Starlikeness conjecture. For every…

Let $\mathbb{H}^2=\{z\in\mathbb{C}:\operatorname{Im}z>0\}$, let $\delta_{\mathbb{H}^2}(x)$ denote the Euclidean distance from $x$ to the boundary, and let…