6 problems

Let $\Omega$ be a simply connected Jordan region in the plane. For $N\in\mathbb N$, let $\rho_N(\Omega)$ denote the squared distance from $\bar z$ to the polynomials of degree at m…

Let $P_k\subset\mathbb{R}^2$ be an open $k$-gon, and let $P_k^*\subset\mathbb{R}^d$ be an open regular $k$-gon. For $k\geq 3$, define the fundamental ratio by…

Vertex attainment conjecture. The maximum

Rectangle inequality. For all points $u,v\in R$,

Let $G_k$ be the regular $k$-gon described in the paper, and let $x,y\in\bigcap_k G_k$ be distinct points. Let $\rho_{G_k}$ and $\delta_{G_k}$ denote the Möbius and hyperbolic metr…

Let $G\subsetneq\mathbb{R}^2$ be a bounded convex polygonal domain, and let $\rho_G$ and $\delta_G$ denote the Möbius and hyperbolic metrics in $G$. The convex polygonal-domain ine…