The arc distortion in QH inner .-uniform (or convex) domains in real Banach spaces

Let D and D. be domains in real Banach spaces of dimension at least 2. The main aim of this paper is to study certain arc distortion properties in the quasihyperbolic metric defined in real Banach spaces. In particular, when D. is a QH inner .-uniform domain with . being a slow (or a convex domain), we investigate the following: For positive constants c,h,C,M, suppose a homeomorphism f: D . D. takes each of the 10-neargeodesics in D to (c, h)-solid in D.. Then f is C-coarsely M-Lipschitz in the quasihyperbolic metric. These are generalizations of the corresponding result obtained recently by Väisälä.