  We prove that a hypothesis of Cassels, Swinnerton-Dyer, recast by Margulis as
statement on the action of the diagonal group $A$ on the space of unimodular
lattices, is equivalent to several assertions about minimal sets for this
action. More generally, for a maximal $\mathbb{R}$-diagonalizable subgroup $A$
of a reductive group $G$ and a lattice $\Gamma$ in $G$, we give a sufficient
condition for a compact $A$-minimal subset $Y$ of $G/\Gamma$ to be of a simple
form, which is also necessary if $G$ is $\mathbb{R}$-split. We also show that
the stabilizer of $Y$ has no nontrivial connected unipotent subgroups.
