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