On the structure of quasiconvex hulls

K Zhang

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23 Citations (Scopus)


We define the set Kq,e ? K of quasiconvex extreme points for compact sets K ? MN X 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 Q(K)q,e ? Kq,e ? [Q(K)q,e]. 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. © Elsevier, Paris.

Original languageEnglish
Pages (from-to)663-686
Number of pages24
JournalAnnales de l'Institut Henri Poincaré (C) Analyse Non Linéaire
Issue number6
Publication statusPublished - Nov 1998


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