Relatively and inner uniform domains

Abstract

We generalize the concept of a uniform domain in Banach spaces into two directions. (1) The ordinary metric d d of a domain is replaced by a metric e ≥ d e\ge d , in particular, by the inner metric of the domain. (2) The uniformity condition is supposed to hold only for certain pairs of points of the domain. We consider neargeodesics and solid arcs in these domains. Applications to the boundary behavior of quasiconformal maps are given. In particular, we study maps between domains of the form E × B E\times B , where E E is a Banach space and B B is a ball.