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On the structure of quasiconvex hulls. (English) Zbl 0917.49014
It is well known that for a compact convex subset $$K$$ of $$\mathbb{R}^n$$ the set $$K_e$$ of all its extreme points is the smallest generator of $$K$$ in the sense that the convex hull $$C(K_e)$$ coincides with $$K$$ and for every compact set $$W\subset K$$ such that $$C(W)= K$$ it is $$K_e\subset W$$.
For a compact subset $$K$$ of $$\mathbb{R}^{N\times n}$$, the author defines the quasiconvex extreme points as the points $$P\in K$$ such that all gradient homogeneous Young measures with support in $$K$$ and barycenter in $$P$$ coincide with the Dirac maps $$\delta_P$$. The set of all quasiconvex extreme points of $$K$$ is denoted by $$K_{q,e}$$.
In the paper under review several properties of the set $$K_{q,e}$$ are investigated. In particular, it is proved that if $$K$$ is compact and quasiconvex, and if $$Q(E)$$ denotes the quasiconvex hull of a set $$E$$, then $$K_{q,e}$$ is the smallest generator of $$K$$ in the sense that $$Q(K_{q,e})= K$$ and $$K_{q,e}\subset W$$ for every $$W\subset K$$ such that $$Q(W)= K$$.
Reviewer: G.Buttazzo (Pisa)

##### MSC:
 49J45 Methods involving semicontinuity and convergence; relaxation
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