  We prove that, for two-component maps, rank-one convexity is equivalent to
quasiconvexity. The essential tool for the proof is a fixed-point argument for
a suitable set-valued map going from one component to the other and preserving
decomposition directions in the $(H_n)$-condition formalism; the existence of a
fixed point ensures that, in addition to keeping decomposition directions,
joint volume fractions are preserved as well. When maps have more than two
components, then fixed points exist for every combination of two components,
but they do not match in general.

Optimization and Control