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On various semiconvex hulls in the calculus of variations. (English) Zbl 0896.49005
Let $$M^{N\times n}$$ be the set of the $$N\times n$$ matrices and $$K\subseteq M^{N\times n}$$. In the paper some relationships among some semiconvex hulls of $$K$$ are studied. The closed lamination convex hull $$L_c(K)$$ of $$K$$, the rank-one convex hull $$R(K)$$ of $$K$$, the quasiconvex hull $$Q(K)$$ of $$K$$, the polyconvex hull $$P(K)$$ of $$K$$, and the convex hull $$C(K)$$ of $$K$$ are recalled and some of their properties are reported. Then it is proved that if $$K$$ is compact and $$L_c(K)\neq C(K)$$, then also $$Q(K)$$ is different from $$C(K)$$. In the case $$N=n=2$$ it is also proved that if $$K$$ is compact and $$L_c(K)\neq C(K)$$ then $$P(K)$$ is not convex. An estimate of the non convexity of $$Q(K)$$ if $$R(K)$$ is not convex is obtained by using quasiconvex envelopes of some distance functions from $$K$$.

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