The following claim is used for proving the quadratic convergence phase of Newton's method on self-concordance functions. I have a very long and non-intuitive proof using Lagrange Multipliers. I suspect that there is a simple proof using tensors. The claim:
Let $a\in R^{3d-2}$ and let $h:( R^d)^3\to R$ be defined by $$h(u,v,w)=\sum_{i=1}^d \sum_{j=1}^d \sum_{k=1}^d a_{i+j+k-2}u_iv_jw_k.$$ Then $$\sup_{||u||=||v||=||w||=1}h(u,v,w)= \sup_{||x||=1}h(x,x,x).$$