  We simplify the construction of projection complexes due to
Bestvina-Bromberg-Fujiwara. To do so, we introduce a sharper version of the
Behrstock inequality, and show that it can always be enforced. Furthermore, we
use the new setup to prove acylindricity results for the action on the
projection complexes. We also treat quasi-trees of metric spaces associated to
projection complexes, and prove an acylindricity criterion in that context as
well.
