We study R2⊕R-separately convex hulls of finite
sets of points in R3, as introduced in
\cite{KirchheimMullerSverak2003}. When R3 is considered as a
certain subset of 3×2 matrices, this notion of convexity corresponds
to rank-one convex convexity Krc. If R3 is identified instead
with a subset of 2×3 matrices, it actually agrees with the quasiconvex
hull, due to a recent result \cite{HarrisKirchheimLin18}.
We introduce "2+1 complexes", which generalize Tn constructions. For a
finite set K, a "2+1K-complex" is a 2+1 complex whose extremal points
belong to K. The "2+1-complex convex hull of K", Kcc, is the union
of all 2+1K-complexes. We prove that Kcc is contained in the 2+1
convex hull Krc.
We also consider outer approximations to 2+1 convexity based in the
locality theorem \cite[4.7]{Kirchheim2003}. Starting with a crude outer
approximation we iteratively chop off "D-prisms". For the examples in
\cite{KirchheimMullerSverak2003}, and many others, this procedure reaches a
"2+1K-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 finite time, but we show that a sequence of outer
approximations built with D-prisms converges to a 2+1K-complex. We
conclude that Krc is always a "2+1K-complex", which has interesting
consequences