  This article is dedicated to geometric structure of the Lorentz and
Marcinkiewicz spaces in case of the pure atomic measure. We study complete
criteria for order continuity, the Fatou property, strict monotonicity and
strict convexity in the sequence Lorentz spaces $\gamma_{p,w}$. Next, we
present a full characterization of extreme points of the unit ball in the
sequence Lorentz space $\gamma_{1,w}$. We also establish a complete description
with an isometry of the dual and predual spaces of the sequence Lorentz spaces
$\gamma_{1,w}$ written in terms of the Marcinkiewicz spaces. Finally, we show a
fundamental application of geometric structure of $\gamma_{1,w}$ to
one-complemented subspaces of $\gamma_{1,w}$.
