  This paper explores some connections between rank one convexity,
multiplicative quasiconvexity and Schur convexity. Theorem 5.1 gives simple
necessary and sufficient conditions for an isotropic objective function to be
rank one convex on the set of matrices with positive determinant. Theorem 6.2
describes a class of possible non-polyconvex but multiplicative quasiconvex
isotropic functions. This class is not contained in a well known theorem of
Ball (6.3 in this paper) which gives sufficient conditions for an isotropic and
objective function to be polyconvex. We show here that there is a new way to
prove directly the quasiconvexity (in the multiplicative form). Relevance of
Schur convexity for the description of rank one convex hulls is explained.
