  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.
