The existence of minimizers in the calculus of variations is closely tied to various notions of convexity. A fundamental question in this regard is how these notions relate to each other. 

In this talk, we study several such generalized notions of convexity for continuous integrands on the space of symmetric $3\times 3$ matrices and establish their mutual relationships.

In particular, we construct a symmetric quasiconvex integrand that is not polyconvex at symmetric matrices, and an integrand that is both symmetric rank-one convex and Hessian quasiconvex but not symmetric quasiconvex. We also establish a transfer principle that converts smooth rank-one convex functions that fail to be Hessian quasiconvex into symmetric rank-one convex integrands while preserving this failure. Applying this scheme to existing counterexamples, we obtain symmetric rank-one convex integrands that are not Hessian quasiconvex. Together with previous results, this settles all pairwise implications among the notions considered here.