  We study the sets that are computable from both halves of some (Martin-L\"of)
random sequence, which we call \emph{$1/2$-bases}. We show that the collection
of such sets forms an ideal in the Turing degrees that is generated by its
c.e.\ elements. It is a proper subideal of the $K$-trivial sets. We
characterise $1/2$-bases as the sets computable from both halves of Chaitin's
$\Omega$, and as the sets that obey the cost function $\mathbf c(x,s) =
\sqrt{\Omega_s - \Omega_x}$.
  Generalising these results yields a dense hierarchy of subideals in the
$K$-trivial degrees: For $k< n$, let $B_{k/n}$ be the collection of sets that
are below any $k$ out of $n$ columns of some random sequence. As before, this
is an ideal generated by its c.e.\ elements and the random sequence in the
definition can always be taken to be $\Omega$. Furthermore, the corresponding
cost function characterisation reveals that $B_{k/n}$ is independent of the
particular representation of the rational $k/n$, and that $B_p$ is properly
contained in $B_q$ for rational numbers $p< q$. These results are proved using
a generalisation of the Loomis--Whitney inequality, which bounds the measure of
an open set in terms of the measures of its projections. The generality allows
us to analyse arbitrary families of orthogonal projections. As it turns out,
these do not give us new subideals of the $K$-trivial sets, we can calculate
from the family which $B_p$ it characterises.
  We finish by showing that the the union of $B_p$ for $p<1$ is the collection
of sets which are robustly computable from a random, a class previously studied
by Hirschfeldt, Jockusch, Kuyper, and Schupp.
