On the Hessian matrix and Minkowski addition of quasiconvex functions

We study the class Q of quasiconvex functions (i.e. functions with convex sublevel sets), by associating to every quasi convex function u a function H :R^n × R→R ∪ {±∞}, such that H(X, t) is nondecreasing in t and sublinear in X: for every fixed t , the function H(·, t) is nothing else than the support function of the sublevel set {x ∈ R^n: u(x)<t}. When u is suitably regular, we establish an exact relation between the Hessian matrices of u and H; this allows us to find explicit formulae to write the k-Hessian operators of u in terms of H. Then we investigate on Minkowski addition of quasiconvex functions.