Hyperbolicity and Vitali properties of unbounded domains in Banach spaces

Let $\Om$ be a unbounded domain in a Banach space. In this work, we wish to impose {\it local conditions} on boundary point of $\Om$ (including the point at infinity) that guarantee complete hyperbolic of $\Om.$ We also search for local boundary conditions so that Vitali properties hold true for $\Om.$ These properties might be considered as analogues of the taut property in the finite dimensional case.