A naive generalization of the hyperbolic and the quasihyperbolic metrics

Although the hyperbolic metric possesses many remarkable properties, it is not defined on arbitrary subdomains of $\mathbb{R}^n$ with $n \geq 2$. This article introduces a new hyperbolic-type metric that provides an alternative approach to this limitation. The proposed metric coincides with the hyperbolic metric on balls and half-spaces, and, quite unexpectedly, agrees with the quasihyperbolic metric in unbounded domains. We compute the density of this metric in several classical domains and discuss aspects of its curvature. Furthermore, we establish characterizations of uniform domains and John disks in terms of the newly defined metric. In addition, we investigate several geometric properties of the metric, including the existence of geodesics and the minimal length of non-trivial closed curves in multiply connected domains.