Quasim\"obius invariance of uniform domains

Abstract

In this paper, we study quasim\"obius invariance of uniform domains in Banach spaces. We first investigate implications of certain geometric properties of domains in Banach spaces, such as the (diameter) uniformity, the $\delta$-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 answer partially to an open question proposed by V\"ais\"al\"a, and provide a new method to prove a recent result of M. Huang, Y. Li, M. Vuorinen, and X. Wang in [On quasim\"obius maps in real Banach spaces, Israel J. Math. 198 (2013), 467-486], which also gives an answer to another question raised by V\"ais\"al\"a.