Kružík, Martin Bauer’s maximum principle and hulls of sets. (English) Zbl 0981.49010 Calc. Var. Partial Differ. Equ. 11, No. 3, 321-332 (2000). Bauer’s maximum principle states that any convex function defined on a compact convex subset of \({\mathbf R}^n\) attains its maximum at some extreme point of this set.In the paper a version of Bauer’s maximum principle is derived for polyconvex, quasiconvex and rank-one convex functions defined on compact subsets of \({\mathbf R}^n\). It is also shown that polyconvex, quasiconvex and rank-one convex hulls of compact sets are generated by polyconvex, quasiconvex and rank-one convex extreme points of the sets. The results are applied to the study of cones of polyconvex, quasiconvex and rank-one convex functions.Some properties of quasiconvex and rank-one convex extreme points are also discussed. In particular, it is shown that they are generally different, and that quasiconvex extreme points are not invariant under the composition with affine mappings which map rank-one matrices into rank-one matrices. Reviewer: Riccardo De Arcangelis (Napoli) Cited in 6 Documents MSC: 49J45 Methods involving semicontinuity and convergence; relaxation 49Q20 Variational problems in a geometric measure-theoretic setting 52A30 Variants of convex sets (star-shaped, (\(m, n\))-convex, etc.) Keywords:quasiconvexity; extreme points; hulls; polyconvexity; rank-one convexity PDFBibTeX XMLCite \textit{M. Kružík}, Calc. Var. Partial Differ. Equ. 11, No. 3, 321--332 (2000; Zbl 0981.49010) Full Text: DOI