On the structure of quasiconvex hulls

Abstract

We define the set $K_{q,e}\subset K$ of quasiconvex extreme points for compact sets $K\subset M^{N\times n}$ 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 $W\subset Q(K)$ with $Q(W)=Q(K)$, $K_{q,e}\subset W$. The set of quasiconvex extreme points relies on $K$ only in the sense that $Q{\left(K\right)}_{q,e}\subset K_{q,e}\subset \overline{[Q{\left(K\right)}_{q,e}]}$. We also establish that $K_e\subset K_{q,e}$, 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.