On the structure of quasiconvex hulls

We define the set K_q,e ? K of quasiconvex extreme points for compact sets K ? M^{N X n} and study its properties. We show that K_q,e is the smallest generator of Q(K)-the quasiconvex hull of K, in the sense that Q(K_q,e) = Q(K), and that for every compact subset W ? Q(K) with Q(W) = Q(K), K_q,e ? W. The set of quasiconvex extreme points relies on K only in the sense that Q(K)_q,e ? K_q,e ? [Q(K)_q,e]. We also establish that K_e ? K_q,e, where K_e is the set of extreme points of C(K)-the convex hull of K. We give various examples to show that K_q,e is not necessarily closed even when Q(K) is not convex; and that for some nonconvex Q(K), K_q,e = K_e. We apply the results to the two well and three well problems studied in martensitic phase transitions. © Elsevier, Paris.