  We study $\mathbb{R}^2\oplus\mathbb{R}$-separately convex hulls of finite
sets of points in $\mathbb{R}^3$, as in KirchheimMullerSverak2003. This notion
of convexity, which we call $2+1$ convexity, corresponds to rank-one convex
convexity, or quasiconvexity, when $\mathbb{R}^3$ is identified with certain
subsets of matrices. We introduce '$2+1$ complexes', which generalize $T_n$
constructions, define the '$2+1$-complex convex hull of a set', and prove that
it is an inner approximation to the $2+1$ convex hull.
  We also consider outer approximations to $2+1$ convexity based in the
locality theorem of rank convexity, by iteratively chopping off '$D$-prisms'.
For many finite sets, this procedure reaches a '$2+1$ $K$-complex' in a finite
number of steps, and thus computes the $2+1$ convex hull. We show examples of
finite sets for which this procedure does not reach the $2+1$ convex hull in a
finite number of steps, but we show that there is always a sequence of outer
approximations built with $D$-prisms that converges to a $2+1$ $K$-complex. We
conclude that $K^{rc}$ is always a '$2+1$ $K$-complex', which has interesting
consequences.
