The Hornich-Hlawka functional inequality for functions with positive differences

We analyze the role played by $n$-convexity for the fulfillment of a series of linear functional inequalities that extend the Hornich-Hlawka functional inequality, $f\left(x\right)+f\left(y\right)+f\left(z\right)+f(x+y+z)\ge f(x+y)+f(y+z)+f(z+x)+f(0),$ including extensions to the case of positive operators.