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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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