Polyconvex, quasiconvex and rank one convex sets

Abstract

We now discuss the notions of polyconvex, quasiconvex and rank one convex sets. Contrary to the usual presentation of classical convex analysis, where the notion of a convex set is defined prior to that of a convex function; this is not the case for the generalized notions of convexity. This is of course due to historical reasons. The notions of polyconvex, quasiconvex and rank one convex functions were introduced, as already said, by Morrey in 1952, although the terminology is that of Ball [53]. It was not until the systematic studies of partial differential equations and inclusions by Dacorogna-Marcellini and Müller-Sverak, initiated in 1996 and discussed in Chapter 10, that the equivalent definitions for sets became an important issue. Moreover these notions were essentially seen through the different generalized convex hulls, leading somehow to terminologies that do not exactly cover the same concepts. We here try to imitate as much as possible the classical approach of convex analysis in the present context. Throughout the two first sections, we follow the presentation of Dacorogna-Ribeiro [213], following earlier results of Dacorogna- Marcellini [202].