3 problems

Sharp quasi-metric constant conjecture. For every $\alpha>0$, $p^\alpha_{\mathbb{B}^n}$ is a quasi-metric with sharp constant $c_*(\alpha)$. Equivalently, $c_*(\alpha)$ is the best…

Let $Z$ be the $z$-axis in $\mathbb{R}^3$, and let $G=\mathbb{R}^3\backslash Z$. The point pair function $p_G$ is defined on pairs of points in $G$. Metricity conjecture. The point…

Metricity conjecture. For a constant $\alpha>0$, this function, defined for $x,y\in\mathbb{R}^n\backslash\overline{\mathbb{B}}^n$ and $n\geq2$, is a metric if and only if…