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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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