Further Study on Domains and Quasihyperbolic Distances

We establish constructive geometric tools for determining when a domain is $L^s$-averaging and obtain upper and lower bounds for the $L^s$-integrals of the quasihyperbolic distance. We also construct examples which are helpful to understand our geometric tools and the relationship between $p$-Poincar\'{e} domains and $L^s$-averaging domains. Finally, finite unions of $L^s(\mu)$-averaging domains are explored.