Polyconvex, quasiconvex and rank one convex envelopes

In Section 6.2, we start with the polyconvex envelope Pf, which is the most similar to the convex envelope Cf. We always recall, without proofs, what has already been said about Cf in Chapter 2 to show the resemblance between the two envelopes. In Section 6.3, we give a representation formula for the quasiconvex envelope, inspired by Carathéodory theorem. In Section 6.4, we discuss a representation formula for Rf, also in the spirit of Carathéodory theorem. In Section 6.5, we present a result that in some cases can simplify the computations of the different envelopes. In Section 6.6, we discuss several examples, relevant for applications, where one can compute these envelopes.