A proper subdomain G of the unit disk \(\mathbb {D}\) is horocyclically convex (horo-convex) if, for every \(\omega \in \mathbb {D}\cap \partial G\) , there exists a horodisk H such that \(\omega \in \partial H\) and \(G\cap H=\emptyset\) . In this paper we give an internal characterization of these domains, namely, that G is horo-convex if and only if any two points can be joined inside G by a \(C^1\) curve composed with finitely many Jordan arcs with hyperbolic curvature in \((-2,2)\) . We also give a lower bound for the hyperbolic metric of horo-convex regions as well as some of its consequences.
