On the quasiconvex exposed points (Q2701819)

The study of quasiconvex sets and hulls arises in the variational approach to martensitic phase transitions and microstructures.NEWLINENEWLINENEWLINEThe author, in his previous paper ``On the structures of quasiconvex hulls'' [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 15, No. 6, 663-686 (1998; Zbl 0917.49014)], introduced the notion of quasiconvex extreme points to answer the question whether there exists a ``smallest'' subset \(K_0\) of a quasiconvex compact set \(K\) of matrices such that \(Q(K_0)=K\), where \(Q(K_0)\) denotes the quasiconvex hull of \(K_0\).NEWLINENEWLINENEWLINEIn the present paper, the author introduces the notion of quasiconvex exposed points for giving a geometric description of quasiconvex extreme points for a compact set of matrices. Moreover, he gives a Straszewicz type theorem, where he proves that for a compact set of matrices the set of quasiconvex exposed points is dense in the set of quasiconvex extreme points.NEWLINENEWLINENEWLINEFurthermore, he analyses the previous notions in examples related to model microstructures.NEWLINENEWLINENEWLINEFinally, he studies the sub-level sets for the explicit quasiconvex relaxation of the squared distance function to a special two point set obtained by Kohn.