Functions with Positive Differences on Convex Cones

We analyze the role played by functions with positive differences defined on convex cones. In particular, we study functions that satisfy linear functional inequalities that extend the three-variable Hornich-Hlawka functional inequality, f (x) + f (y) + f (z) + f (x + y + z) = f (x + y) + f (y + z) + f (z + x) + f (0), especially to the case of n variables. Beyond the classical setting, we present extensions to the case of positive operators.