A note on equality of functional envelopes. (English) Zbl 1028.49007

Kružík, Martin

Math. Bohem. 128, No. 2, 169-178 (2003).

Summary: We characterize generalized extreme points of compact convex sets. In particular, we show that if the polyconvex hull is convex in \(\mathbb R^{m \times n}\), \(\min (m,n)\leq 2\), then it is constructed from polyconvex extreme points via sequential lamination. Further, we give theorems ensuring equality of the quasiconvex (polyconvex) and the rank-1 convex envelopes of a lower semicontinuous function without explicit convexity assumptions on the quasiconvex (polyconvex) envelope.

MSC:

49J45	Methods involving semicontinuity and convergence; relaxation
52A20	Convex sets in \(n\) dimensions (including convex hypersurfaces)

Keywords:

extreme points; polyconvexity; quasiconvexity; rank-1 convexity; lower semicontinuous function