Väisälä's uniqueness, prolongation and convexity conjectures for quasihyperbolic balls

Let $\Omega \subset \mathbb{R}^n$ be a domain and let $x,y \in \Omega$. For $z\in\Omega$, write $d(z)=\operatorname{dist}(z,\partial\Omega)$, and for a rectifiable curve $\gamma$ in $\Omega$ define its quasihyperbolic length by

$$\ell_k(\gamma)=\int_\gamma \frac{|dz|}{d(z)}.$$

The quasihyperbolic metric is $k(x,y)=\inf_\gamma\ell_k(\gamma)$, where the infimum is over curves joining $x$ to $y$, and $B_k(x,r)=\{z\in\Omega:k(x,z)<r\}$.

Väisälä's uniqueness, prolongation and convexity conjectures. There are universal constants $c_u,c_p,c_c>0$ such that:

1. if $k(x,y)<c_u$, then there is only one quasihyperbolic geodesic from $x$ to $y$;
2. if $\gamma$ is a quasihyperbolic geodesic from $x$ to $y$ with $\ell_k(\gamma)=k(x,y)<c_p$, then there is a quasihyperbolic geodesic $\gamma'$ from $x$ to some $y'$ such that $\gamma\subset\gamma'$ and $\ell(\gamma')=c_p$;
3. the quasihyperbolic ball $B_k(x,r)$ is strictly convex for all $x\in\Omega$ and $r<c_c$.

These conjectures concern local uniqueness and extension of quasihyperbolic geodesics and the local convexity of quasihyperbolic metric balls. The surrounding discussion notes that quasihyperbolic geodesics exist in every domain and that quasihyperbolic balls are therefore connected, while smoothness and stronger geometric properties require additional hypotheses; the status of these three conjectures is not specified here.