Constantin Niculescu
University of Craiova, Craiova, Romania

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, $$f(x)+f(y)+f(z)+f(x+y+z)\geq f(x+y)+f(y+z)+f(z+x)+f(0).$$