# 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: silhavy@math.cas.cz

Summary:

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