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)