The theory of outer $L^p$ spaces and the associated quasi-norms were
introduced by Do and Thiele as a framework to prove the boundedness of
operators both in Calderón.-Zygmund theory and time-frequency analysis.
Accordingly to this purpose, the theory was developed in the direction of
the real interpolation features of these spaces, while other questions
remained untouched. In this talk we prove that the expected duality between
outer $L^p$ spaces and a countable triangle inequality for the associated
quasi-norms hold true for p>1 uniformly in the finite setting, while for p=1
we exhibit a counterexample to uniformity. Moreover, we show that in the
upper half space setting the desired properties hold true up to the endpoint
p=1. As a consequence, we establish the equivalence of the outer L^p spaces
in this setting with the classical tent spaces. The main tool in proving the
desired results is a greedy decomposition of functions in the outer $L^p$
spaces that isolate the relevant part of the functions on disjoint sets.