The Hessian matrix of Lagrange function

We study the Hessian matrix of the Lagrange function of a dense 3-uniform hypergraph and show that the Lagrange function of a dense 3-uniform hypergraph has a unique optimal weight. We also give a characterization to the optimal weight of a dense 3-uniform hypergraph by the Hessian matrix of the Lagrange function and a simple application on the regular 3-uniform hypergraphs.