  We show weak lower semi-continuity of functionals assuming the new notion of
a "convexly constrained" $\mathcal A$-quasiconvex integrand.
  We assume $\mathcal A$-quasiconvexity only for functions defined on a set $K$
which is convex. Assuming this and sufficient integrability of the sequence we
show that the functional is still (sequentially) weakly lower semi-continuous
along weakly convergent "convexly constrained" $\mathcal A$-free sequences. In
a motivating example, the integrand is $-\det^{\frac{1}{d-1}}$ and the convex
constraint is positive semi-definiteness of a matrix field.
