Polyconvex, quasiconvex and rank one convex functions

We should again emphasize that in the scalar case all these notions are equivalent to the usual convexity condition. The definitions and main properties of these generalized notions of convexity are discussed in Section 5.2. In Section 5.3, we give several examples. In particular we show that all the reverse implications are false. Finally, in an appendix (Section 5.4), we gather certain elementary properties of determinants.