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

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 $\xi_{0}$