Bauer's maximum principle and hulls of sets

Abstract

We use the Bauer maximum principle for quasiconvex, polyconvex and rank-one convex functions to derive Krein-Milman-type theorems for compact sets in ${\mathbb R}^{m\times n}$ . Further we show that in general the set of quasiconvex extreme points is not invariant under transposition and it is different from the set of rank-one convex extreme points. Finally, a set in ${\mathbb R}^{3\times 3}$ with different polyconvex, quasiconvex and rank-one convex hulls is constructed.