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On the structure of quasiconvex hulls

By Kewei Zhang


We define the set Kq,e ⊂ K of quasiconvex extreme points for compact sets K ⊂ MN×n and study its properties. We show that Kq,e is the smallest generator of Q(K)-the quasiconvex hull of K, in the sense that Q(Kq,e) = Q(K), and that for every compact subset W ⊂ Q(K) with Q(W) = Q(K), Kq,e ⊂ W. The set of quasiconvex extreme points relies on K only in the sense that View the MathML source. We also establish that Ke ⊂ Kq,e, where Ke is the set of extreme points of C(K)-the convex hull of K. We give various examples to show that Kq,e is not necessarily closed even when Q(K) is not convex; and that for some nonconvex Q(K), Kq,e = Ke. We apply the results to the two well and three well problems studied in martensitic phase transitions

Publisher: Elsevier
Year: 1998
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