  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, $f\left( x\right) +f\left( y\right) +f\left( z\right) +f\left(
x+y+z\right) \geq f\left( x+y\right) +f\left( y+z\right)+f\left( z+x\right)
+f(0),$ including extensions to the case of positive operators.
