What, if any, notions of positive definiteness can be extended to 3rd order tensors (and beyond)?
The reason I ask is because the Hessian matrix of a convex function is positive semi-definite, but trying to evaluate the convexity of matrix valued functions can occasionally result in Hessians that are 3rd order (or greater) tensors. Is there an analog of positive definiteness for such objects? And even if there is, would such a property holding imply that the original function was convex?