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Journal of Convex Analysis 18 (2011), No. 1, 203--208
Copyright Heldermann Verlag 2011

A Remark on the Structure of the Busemann Representative of a Polyconvex Function

Jon J. Bevan
Dept. of Mathematics, University of Surrey, Guildford, GU2 7XH, United Kingdom


\newcommand{\R}{{\bf R}} Under mild conditions on a polyconvex function $W: \R^{2 \times 2} \to \R$, its largest convex representative, known as the Busemann representative, may be written as the supremum over all affine functions $\phi: \R^{5} \to \R$ satisfying $\phi(\xi,\det \xi) \leq W(\xi)$ for all $ 2 \times 2$ matrices $\xi$. In this paper, we construct an example of a polyconvex $W: \R^{2 \times 2} \to \R$ whose Busemann representative is, on an open set, strictly larger than the supremum of all affine functions $\phi$ as above and which also satisfy $\phi(\xi_{0},\det \xi_{0}) = W(\xi_{0})$ for at least one $2 \times 2$ matrix $\xi_{0}$.

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