Preprint no. 150
Mathematical Institute of the Academy of Sciences of the Czech Republic

An O(n) invariant quasiconvex function that is not polyconvex

Miroslav Silhavy

Mathematical Institute, Acad. Sci.of the Czech Republic, 115 67 PRAHA 1, Zitna 25, Czech Republic, e-mail:


It is noted that the quasiconvex hull of the euclidean distance from a compact nonconvex quasiconvex $O(n)$ invariant subset of the space $M^{n\times n}$ of $n\times n$ real matrices is not polyconvex. This provides an example of an $O(n)$ invariant quasiconvex (and hence rank 1 convex) function that is not polyconvex and answers a recent question by Dacorogna & Marcellini and Cardaliaguet & Tahraoui. 

Keywords: quasiconvexity, polyconvexity, rank 1 convexity

Mathematics Subject Classification 2000: 49J45

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