Embeddings of concave functions and duals of Lorentz spaces

A simple expression is presented that is equivalent to the norm of the $L^p_v\to L^q_u$ embedding of the cone of quasi-concave functions in the case $0 < q < p < \infty$. The result is extended to more general cones and the case $q=1$ is used to prove a reduction principle which shows that questions of boundedness of operators on these cones may be reduced to the boundedness of related operators on whole spaces. An equivalent norm for the dual of the Lorentz space
$$
\Gamma_p(v) = \left\{ f: \left( \int_0^\infty (f^{**})^pv \right)^{1/p} < \infty \right\}
$$
is also given. The expression is simple and concrete. An application is made to describe the weights for which the Hardy Littlewood Maximal Function is bounded on these Lorentz spaces.