On the different notions of convexity for rotationally invariant functions (Q1327529)

The authors consider functions \(f: M^{2\times 2}\to \mathbb{R}\) (where \(M^{2\times 2}\) denotes all real \(2\times 2\) matrices), which are invariant in the sense that \(f(RXQ)= f(X)\) for each \(R, Q\in SO(2)\). Such functions are determined by their values on the set \({\mathcal D}\subset M^{2\times 2}\) of the diagonal matrices. The authors consider various convexity conditions introduced in the calculus of variations (namely rank-one convexity, quasi-convexity, polyconvexity and the usual convexity; see, for example, the paper of \textit{J. Ball} [Arch. Ration. Mech. Anal. 63, 337-403 (1977; Zbl 0368.73040)]) and investigate the question to what extent the convexity conditions which are satisfied by the restriction of \(f\) to \(\mathcal D\) imply the convexity conditions for the whole function \(f: M^{2\times 2}\to \mathbb{R}\). The main result is that convexity (respectively polyconvexity) of the restriction \(f: {\mathcal D}\to \mathbb{R}\) implies the convexity (respectively polyconvexity) of the full function \(f: M^{2\times 2}\to \mathbb{R}\), whereas this fails for the other two convexity conditions above.