On the Hessian matrix and Minkowski addition of quasiconvex functions

AbstractWe study the class Q of quasiconvex functions (i.e. functions with convex sublevel sets), by associating to every u∈Q∩C(Rn) a function H:Rn×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∈Rn:u(x)⩽t}. When u is suitably regular, we establish an exact relation between D2u and D2H; this allows us to find explicit formulae to write the k-Hessian operators Sk(D2u) (among which Δu and detD2u) in terms of H. Then we investigate on Minkowski addition of quasiconvex functions