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On some semiconvex envelopes. (English) Zbl 1012.49012
Let $$M^{N\times n}$$ denote the set of the ($$N\times n$$)-matrices with real entries. For every $$f\colon M^{N\times n}\to{\mathbb R}$$ let $$Cf$$, $$Qf$$, and $$Rf$$ denote, respectively, the convex, quasiconvex, rank-one convex envelopes of $$f$$. Then $Cf\leq Qf\leq Rf\leq f.$ In the paper it is proved that, if $$f$$ is superlinear in the sense that $\lim_{|A|\to\infty}{f(A)\over|A|}=+\infty,$ then $$Cf=Qf$$ if and only if $$Cf=Rf$$.
I particular, if a rank-one convex superlinear function is not convex, then its quasiconvex envelope is not convex. This remark provides a way for testing whether the quasiconvex envelope of a superlinear function is trivial (i.e., convex) by just calculating its rank-one convex envelope.

##### MSC:
 49J45 Methods involving semicontinuity and convergence; relaxation 49J10 Existence theories for free problems in two or more independent variables 52A40 Inequalities and extremum problems involving convexity in convex geometry
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