Quasimöbius invariance of uniform domains
We study quasimöbius invariance of uniform domains in Banach spaces. We first investigate implications of certain geometric properties of domains in Banach spaces, such as (diameter) uniformity, $\delta $-uniformity and the min-max property. Then we show that all of these conditions are equivalent if the domain is $\psi $-natural. As applications, we partially answer an open question proposed by Väisälä, and provide a new method to prove a recent result of Huang et al. (2013), which also gives an answer to another question raised by Väisälä.