The two-well problem in three dimensions

We study properties of generalized convex hulls of the set K = SO(3) ∪ SO(3)H with det H > 0. If K contains no rank-1 connection we show that the quasiconvex hull of K is trivial if H belongs to a certain (large) neighbourhood of the identity. We also show that the polyconvex hull of K can be nontrivial if H is sufficiently far from the identity, while the (functional) rank-1 convex hull is always trivial. If the second well is replaced by a point then the polyconvex hull is trivial provided that there are no rank-1 connections.