Characterization of polyconvex isotropic functions

Polyconvexity is an important concept in the analysis of energies related to elasticity. A function $f \colon \R^{d\times d} \to \R$ is called polyconvex if it can be written as a convex function in the minors of the argument. We show that for isotropic functions it suffices to consider diagonal matrices. For $d=3$, this leads to a dimension reduction for the convex representative of $f$ from $\R^{19}$ to $\R^7$. Moreover, we present a new result for the polyconvexity of functions formulated in the principal invariant of the left or right stretch tensor.